Absorption and refraction, drawn as one function

At its defaults it draws absorption and refraction, drawn as one function. The real and imaginary parts of a Lorentz oscillator's susceptibility against frequency, in units of the resonance. The imaginary part is the absorption: a symmetric line centred on the resonance, with a full width at half maximum equal to the damping — 0.05, 0.12, 0.30 here. The real part is the refraction, and it is what the same medium does to the speed of light. The two curves are not two facts about the medium: either one determines the other completely, by an integral over all frequencies, and that is a consequence of the medium responding after it is asked rather than before. Between x = 0.97 and x = 1.02 the refraction runs the wrong way — the index falls as the frequency rises, which is anomalous dispersion — and that region is exactly the width of the absorption line. Away from the line the index rises with frequency, which is ordinary dispersion and is why a prism separates colours in the order it does: every transparent material is on the low-frequency tail of an ultraviolet absorption it is not otherwise showing. A narrower line is a taller one, because the area under the absorption is fixed by how many electrons there are and by nothing else.

kramers-kronig is one function in lib/figures/waves.js — travelling, standing, adding and shifting. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

The real and imaginary parts of a Lorentz oscillator's susceptibility against frequency, in units of the resonance. The imaginary part is the absorption: a symmetric line centred on the resonance, with a full width at half maximum equal to the damping — 0.05, 0.12, 0.30 here. The real part is the refraction, and it is what the same medium does to the speed of light. The two curves are not two facts about the medium: either one determines the other completely, by an integral over all frequencies, and that is a consequence of the medium responding after it is asked rather than before. Between x = 0.97 and x = 1.02 the refraction runs the wrong way — the index falls as the frequency rises, which is anomalous dispersion — and that region is exactly the width of the absorption line. Away from the line the index rises with frequency, which is ordinary dispersion and is why a prism separates colours in the order it does: every transparent material is on the low-frequency tail of an ultraviolet absorption it is not otherwise showing. A narrower line is a taller one, because the area under the absorption is fixed by how many electrons there are and by nothing else.

The real and imaginary parts of a Lorentz oscillator's susceptibility against frequency, in units of the resonance. The imaginary part is the absorption: a symmetric line centred on the resonance, with a full width at half maximum equal to the damping — 0.05, 0.12, 0.30 here. The real part is the refraction, and it is what the same medium does to the speed of light. The two curves are not two facts about the medium: either one determines the other completely, by an integral over all frequencies, and that is a consequence of the medium responding after it is asked rather than before. Between x = 0.97 and x = 1.02 the refraction runs the wrong way — the index falls as the frequency rises, which is anomalous dispersion — and that region is exactly the width of the absorption line. Away from the line the index rises with frequency, which is ordinary dispersion and is why a prism separates colours in the order it does: every transparent material is on the low-frequency tail of an ultraviolet absorption it is not otherwise showing. A narrower line is a taller one, because the area under the absorption is fixed by how many electrons there are and by nothing else.

The response of the medium to a sharp kick at time zero, computed two ways. The solid curve is the true response: nothing at all before the kick, then a damped ringing at the resonance. The second curve is what comes back if the medium is given the same absorption and no dispersion — the imaginary part kept and the real part set to zero, which is what quoting an absorption spectrum with a constant refractive index amounts to. Half the response moves to before the kick. It is exactly half and it is exactly mirrored, because keeping only the odd part of a function keeps half of it on each side; the largest excursion before t = 0 is 0.274 against a peak of 0.547. That is the whole argument for the Kramers–Kronig relations in one picture: the constraint between absorption and refraction is not a property of oscillators, or of light, or of any model. It is the requirement that the answer come after the question, and a medium whose two halves are chosen independently answers first.

The absorption line is the only input. The dotted curve is the refractive part computed from it by the Kramers–Kronig integral — a principal value over all frequencies, evaluated here by subtraction so that no window has to be excised around the pole — and the solid curve is the exact real part of the same oscillator. They agree to 2.4e-7, which is better than a thousandth of the oscillator strength. Nothing about the shape of the refraction was assumed: the computation is handed a symmetric absorption line and returns an antisymmetric dispersion curve with the anomalous region in the right place and the right depth. The content of that agreement is that a medium has one degree of freedom per frequency and not two. Measure how much light a material absorbs at every wavelength and its refractive index at every wavelength is decided; measure the index everywhere and the absorption is decided. The integral is the reason an absorption at 100 nm sets the index at 600 nm, so the dispersion of a transparent glass is a statement about the ultraviolet lines it does not show in the visible.

A medium with 3 absorption lines, drawn as the sum of their contributions. The area under the absorption weighted by frequency is fixed: the numerical integral comes to 2.771 against the exact π/2 times the total oscillator strength, 2.771, a relative difference of 3.0e-5. Widening a line lowers it by exactly as much as it broadens it, and moving a line changes nothing at all: the total is a count of the electrons available to be driven and of nothing else, which is why it is called a sum rule. The consequence is on the other curve. Above every resonance the susceptibility is negative — the electrons are driven out of step with the field — so the refractive index is less than one, crossing unity at 2.08 times the first resonance here. That is not a violation of anything, because the speed a signal travels is not the phase speed; it is the reason X-rays refract the wrong way, why an X-ray mirror works only at grazing incidence, and why the refractive index of every material tends to one from below as the frequency rises past the last electron it has.

The real and imaginary parts of a Lorentz oscillator's susceptibility against frequency, in units of the resonance. The imaginary part is the absorption: a symmetric line centred on the resonance, with a full width at half maximum equal to the damping — 0.30 here. The real part is the refraction, and it is what the same medium does to the speed of light. The two curves are not two facts about the medium: either one determines the other completely, by an integral over all frequencies, and that is a consequence of the medium responding after it is asked rather than before. Between x = 0.84 and x = 1.14 the refraction runs the wrong way — the index falls as the frequency rises, which is anomalous dispersion — and that region is exactly the width of the absorption line. Away from the line the index rises with frequency, which is ordinary dispersion and is why a prism separates colours in the order it does: every transparent material is on the low-frequency tail of an ultraviolet absorption it is not otherwise showing. A narrower line is a taller one, because the area under the absorption is fixed by how many electrons there are and by nothing else.

physicscheck asserts something about kramers-kronig that could fail — it draws it and measures the result against a value reached some other way.

Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.