Why a Focused Laser Can Double the Frequency of Light — Epoche C1
In 1961 a pulse of red light from a ruby laser passed through a slab of crystalline quartz and a faint trace of ultraviolet, at exactly half the incoming wavelength, emerged on the far side. Nothing comparable happens when two torch beams cross, and textbooks list the refractive index — the factor by which a material slows light — as a constant of the material. Both of those familiar facts are approximations, and both fail for the same quantitative reason: the electric field of ordinary light is minute beside the field that binds an electron to its atom. This essay traces the chain from that ratio to Franken's ultraviolet spot, and then to the phase-matching condition that separates a curiosity converting one photon in a hundred million from the doubling crystal inside a green laser pointer. The ratio that makes optics linear The whole argument rests on comparing two electric fields, so this section constructs both and divides them. The first is the field an electron feels inside an atom. Its natural yardstick is the proton's Coulomb field at one Bohr radius $a_0 = 5.3\times10^{-11}\,\mathrm{m}$, the orbital radius of hydrogen in Bohr's model: $$E_{\mathrm{at}} = \frac{e}{4\pi\varepsilon_0 a_0^2} \approx 5\times10^{11}\,\mathrm{V\,m^{-1}},$$ with $e$ the elementary charge and $\varepsilon_0$ the permittivity of free space (SI units throughout). The second field is that of the light. For a plane wave the time-averaged energy flux — the Poynting flux, energy crossing unit area per second — is $I = \tfrac{1}{2}\varepsilon_0 c E_0^2$, where $E_0$ is the amplitude of the oscillating field and $c$ the speed of light. Bright sunlight delivers about $1\,\mathrm{kW\,m^{-2}}$, so $E_0 = \sqrt{2I/\varepsilon_0 c} \approx 9\times10^{2}\,\mathrm{V\,m^{-1}}$. Sunlight therefore sits almost nine orders of magnitude below the atomic field. That ratio is not merely suggestive; it is exactly the small parameter of the expansion, and a two-line model shows why. Treat a bound electron as a particle of mass $m$ in a smooth potential well, so that for small displacements $x$ from equilibrium the restoring force is $-m\omega_0^2 x$ plus corrections. The stiffness is set by the same Coulomb attraction as $E_{\mathrm{at}}$, giving $m\omega_0^2 \approx e^2/(4\pi\varepsilon_0 a_0^3)$. A driving field $E$ well below resonance displaces the electron by $x \approx eE/(m\omega_0^2)$, and substituting the stiffness gives $$x \approx \frac{eE\,4\pi\varepsilon_0 a_0^3}{e^2} = a_0\,\frac{E}{E_{\mathrm{at}}}.$$ The field ratio is the fractional displacement of the electron within its orbit. In sunlight the electron is pushed about two parts in $10^{9}$ of the way across the atom, so it never leaves the region where the potential is well approximated by a parabola, the restoring force stays proportional to the displacement, and the induced polarisation $P$ — the dipole moment per unit volume — follows the field faithfully. Writing the response as a power series in the instantaneous field, $$P = \varepsilon_0\left(\chi^{(1)}E + \chi^{(2)}E^2 + \chi^{(3)}E^3 + \cdots\right),$$ where $\chi^{(1)}$ is the linear susceptibility that fixes the refractive index through $n = \sqrt{1+\chi^{(1)}}$, the same estimate fixes the size of the higher coefficients. Since the expansion parameter of the anharmonic terms is $x/a_0 = E/E_{\mathrm{at}}$, each coefficient is smaller than its predecessor by roughly one power of the atomic field: $\chi^{(2)} \sim \chi^{(1)}/E_{\mathrm{at}}$, $\chi^{(3)} \sim \chi^{(1)}/E_{\mathrm{at}}^2$. In sunlight the quadratic term is suppressed by $2\times10^{-9}$ relative to the linear one. The response is linear, superposition holds, and beams pass through one another untouched. Two caveats on the series belong here rather than later: it presumes the response is instantaneous, which holds only far from any absorption resonance, and it presumes a scalar relation between two vectors, which the next section but one has to abandon. What a focused laser changes The laser did not create a new physical effect; it moved $E_0$ up the scale just built. Maiman (1960) obtained pulsed coherent emission at $694.3\,\mathrm{nm}$ from a ruby rod, and the instrument Franken, Hill, Peters and Weinreich (1961) used delivered about $3\,\mathrm{J}$ in a pulse of millisecond order — a peak power near $3\,\mathrm{kW}$. Coherence is what makes focusing possible: the beam can be brought to a spot of order a tenth of a millimetre across, an area near $8\times10^{-9}\,\mathrm{m^2}$, giving an intensity around $4\times10^{11}\,\mathrm{W\,m^{-2}}$ and, by the flux formula above, $E_0 \approx 2\times10^{7}\,\mathrm{V\,m^{-1}}$. That is $4\times10^{-5}$ of the atomic field — still small, but five orders of magnitude closer than sunlight. At that level the quadratic term does something the linear term cannot: it manufactures a frequency that was not present. Write the driving field as $E = E_0\cos\omega t$, with $\omega$ the angular frequency. Then $$E^2 = E_0^2\cos^2\omega t = \tfrac{1}{2}E_0^2\left(1+\cos 2\omega t\right).$$ The quadratic polarisation therefore has two parts: a constant term, which is a static polarisation of the crystal — optical rectification, detected in the same laboratory the following year by Bass, Franken, Ward and Weinreich (1962) — and a term oscillating at twice the input frequency. An oscillating polarisation is a dense array of oscillating dipoles, and oscillating dipoles radiate. That is the mechanism: red light in, ultraviolet out. In the 1961 experiment $694.3\,\mathrm{nm}$ entered quartz and $347.2\,\mathrm{nm}$ emerged. The order of magnitude also fits, though the agreement should not be oversold. The harmonic amplitude relative to the fundamental is of order $E_0/E_{\mathrm{at}}$, so the converted intensity fraction is of order $(4\times10^{-5})^2 \approx 2\times10^{-9}$. A fundamental photon carries $hc/\lambda \approx 2.9\times10^{-19}\,\mathrm{J}$ ($h$ is Planck's constant, $\lambda = 694\,\mathrm{nm}$