The Carvallo paradox and femtosecond lasers - part I

Fourier analysis amounts to writing any signal (including those limited in time) as a sum of infinitely extended harmonic functions. One way of framing this apparent contradiction is the Carvallo paradox:

Since a spectrograph only selects one component of the signal to be analyzed (and this component is infinite in time), it should detect that component both before and after receiving the signal.

The standard answer is that spectrographs have a finite resolution: when selecting light with a given wavelength \(\lambda\), the result is in practice a finite interval \( ( \lambda - \Delta \lambda,  \lambda + \Delta \lambda)\), defining the spectral resolution \( R = \lambda / \Delta \lambda \). Let us assume a (respectable) value \( R = 10^5\) in the visible range, at \(\lambda = 500 \, \text{nm}\).

The uncertainty relation connects the resolution to the lenth of the pulse by: \[ \Delta \omega \Delta t \geq \frac{1}{2} \Rightarrow \Delta t = \frac{R}{2 \omega} \geq 8 \; 10^{-11} \, \text{s} \simeq 100 \, \text{ps}\] where in the second step I used \( \left | \Delta \omega / \omega \right | = \left | \Delta \lambda / \lambda \right |\). A pulse with the spectral sharpness \(\Delta \lambda\) must then be at least 100 ps long.

This is a very short interval for classical spectroscopy measurements. However, using modern optical techniques one can create ultrashort pulses, down to \( \Delta t \simeq 10 \, \text{fs}\). When observing such a pulse at a resolution \( R \) we should then see it as spread out to 100 ps. Where is the error?

More about in in the next part, where I'll also try to give a version of the paradox that is not affected by resolution.

Gaussian integral of an error function

In Surely You're Joking, Mr. Feynman!, Richard Feynman mentions a useful technique he used for evaluating integrals, namely taking the derivative under the integral sign. I will show here how this trick works in calculating the Gaussian integral of an error function. Averages over Gaussian distributions are omnipresent in physics, and the error function is just the primitive of the Gaussian, making the calculations relatively easy (and the result quite elegant.) Nevertheless, Mathematica (version 8) cannot perform this integral, and I could not find it in Gradshteyn & Ryzhik. I needed it to describe the interaction of a phase front with an external field, see the paper here.

Let us define: \[ I(\alpha, \beta, \gamma ) = \int_{-\infty}^{\infty} \text{d}x \exp (-\alpha x^2) \,\text{erf}(\beta x + \gamma) \] with \( \alpha, \beta \, \text{and}\, \gamma\) real and \( \alpha \) positive. For \( \gamma = 0\) the integrand is an odd function, so \( I(\alpha, \beta, 0 ) = 0\). We can also estimate \[ I' (\gamma) = \frac{\partial}{\partial \gamma} I(\alpha, \beta, \gamma) = \frac{2}{\sqrt{\pi}} \int_{-\infty}^{\infty} \text{d}x \exp (-\alpha x^2)\, \exp \left [-(\beta x + \gamma)^2\right ] \] which is a simple Gaussian integral: \( \displaystyle I' (\gamma) = \frac{2}{\sqrt{\alpha + \beta^2}} \exp \left ( - \frac{\alpha \gamma ^2}{\alpha + \beta^2}\right )\)

Finally, \[ I(\alpha, \beta, \gamma ) = \int_{0}^{\gamma} \text{d}u \, I' (u) = \sqrt{\vphantom{\beta}\frac{\pi}{\alpha}} \,\text{erf} \; \left ( \gamma \sqrt{\frac{\vphantom{\beta} \alpha}{\alpha + \beta^2}} \right )\] The reader can check that all derivatives exist and all integrals converge. What happens if we replace the linear term in the error function by a quadratic one?

An aesthetic argument against solipsism

In his short story The Other, from the volume The Book of Sand, Borges uses an artistic proof to convince his interlocutor (his younger self, actually) that their encounter is real and not a mere dream. He quotes a verse from Hugo (that the young Borges had not yet read), which is so striking that it could not have been dreamed up; the two then communicate across half a century.

This being a Borges story, things are more complicated than the summary above. In particular, the older Borges concludes that, although they had met, the younger one had in fact been dreaming. What I am interested in here is whether the story puts forward a successful argument against solipsism.

Such an argument requires demonstrating the existence of another person, who is:

• essentially similar, enough to achieve meaningful communication, and
• sufficiently different, so that it cannot be a figment of the subject's imagination

Does a successful encounter with a work of art fulfill these two conditions?

DIC microscopy image

The edge of a spin-coated droplet (reflection DIC image).

Physics and Photography

Short review of a short but very well-written introduction to the principles of photography: Science for the Curious Photographer: An Introduction to the Science of Photography by Charles S. Johnson Jr (author's blog is here). This slim (180 p.) volume covers a lot of ground, not only the obvious topics such as image formation or the physiology of vision but also, for instance, the operation of CMOS and CCD detectors and a discussion of polarization. The presentation is fairly technical, so an adequate background in physics is necessary.

The only critique would be that, for an introduction, it is somewhat light on the references (many of which are simply Wikipedia articles).

Electrical circuits and Euler's polyhedron formula

When solving an electrical circuit consisting of impedances and voltage sources, one needs to apply Kirchhoff's laws:

• The current law, yielding N-1 equations (with N the number of nodes)
•  The voltage law, for an additional L equations (where L is the number of elementary loops)

The unknowns are the currents flowing in each branch (B of them). We also assume that the branches do not cross. Note that writing the current law for the N-th node or the voltage law for a composite loop (consisting of several adjacent elementary loops) does not provide any further information, the resulting equations being linear combinations of the previous ones.

For the problem to be well-posed the number of unknowns and equations is equal, which we can write as:
\[ N + L - B = 1\] This relation is easily proven in plane geometry, but here I would like to show its intimate connection with Euler's formula, which states that, for a convex polyhedron, \[ V + F - E = 2\] where V, F and E are the numbers of vertices, faces and edges, respectively.

Let us start by establishing a correspondence between circuits and polyhedra, as shown in the figure below. Place a sphere on top of the (planar) circuit diagram and connect each node to the North pole by a line segment (this is known as a stereographic projection.) We define the vertices as the intersections of these segments with the sphere; the result is a convex polyhedron.

It is easily seen that, with the notations above, we have the straightforward equivalences V = N and E = B. The number of faces, however, F = L + 1, since the "topmost" face corresponds to the open area surrounding the circuit. Substitution in either of the equations above yields the other one.

Polarized microscopy image

Schlieren texture in nematic 5CB.