Introduction to Nonlinear Optical Polarization

For a century optics rested on one load-bearing assumption: that a medium's polarization tracks the light field in exact proportion, $P=\varepsilon_0\chi^{(1)}E$ . Every familiar phenomenon — refraction, dispersion, absorption, birefringence — descends from that single linear line, and colours pass through glass without ever breeding new ones. The assumption is superb because ordinary light is feeble on the atomic scale: the field binding a valence electron to its nucleus is of order $E_{at}\approx 5.14\times 10^{11}\,\mathrm{V/m}$ , and even a bright beam sits a fraction of a millionth of the way toward it — too weak to bend the spring it rides on. A focused laser broke that peace. When the driving field becomes an appreciable fraction of $E_{at}$ , the electron is pushed into the anharmonic shoulders of its binding potential, the polarization stops following the field faithfully, and $P$ must be written as a power series, $P=\varepsilon_0(\chi^{(1)}E+\chi^{(2)}E^2+\chi^{(3)}E^3+\cdots)$ . The higher coefficients $\chi^{(2)},\chi^{(3)},\dots$ are the nonlinear susceptibilities, and the moment the series has more than its first term the medium gains a startling new power: because each term is a product of fields, it oscillates at sums and differences of the input frequencies — one colour in, a comb of new colours out. This framework is the trunk from which every branch of nonlinear optics grows: the second-harmonic generation and $\chi^{(2)}$ three-wave mixing of the companion article, the optical Kerr effect and self-focusing beyond it, parametric amplification, entangled-photon sources. Here we set up the language they all speak — the expansion itself, the meaning and size of the $\chi^{(n)}$ , the symmetry rule that forbids even orders in most materials, and the frequency bookkeeping that turns a squared field into a new wave.

Key takeaways

At low fields the polarization is linear, $P=\varepsilon_0\chi^{(1)}E$ ; at laser intensities it extends to a power series $P=\varepsilon_0(\chi^{(1)}E+\chi^{(2)}E^2+\chi^{(3)}E^3+\cdots)$ whose coefficients are the nonlinear susceptibilities.
The yardstick is the atomic unit of electric field, $E_{\rm at}\approx5.14\times10^{11}\,\mathrm{V/m}$ : for a bound electron this sets a characteristic atomic field scale, so higher-order terms are often suppressed by powers of the dimensionless field strength $E/E_{\rm at}$ — although the actual nonlinear response depends strongly on the material and its resonances.
Each nonlinear term is a product of fields, radiating at sums and differences of the inputs: $E^2$ makes DC (rectification) and $2\omega$ ; $E^3$ makes a Kerr correction at $\omega$ and $3\omega$ .
Inversion symmetry is a kill switch for electric-dipole even-order response: in a centrosymmetric medium, $\chi^{(2)}$ and all other even-order electric-dipole susceptibilities vanish, leaving $\chi^{(3)}$ as the leading nonlinearity.
A one-line estimate $\chi^{(2)}\sim\chi^{(1)}/E_{at}\approx 1.9\,\mathrm{pm/V}$ lands within a factor of two of crystals like KDP — right order of magnitude from pure scaling.
Key idea: expand the polarization as a power series, $P=\varepsilon_0\sum_n\chi^{(n)}E^n$ ; each nonlinear term is a product of fields and radiates at new frequencies set by sums and differences of the inputs, with even orders switched off wherever the medium has inversion symmetry.

The core insight

Expand the polarization as a power series in the field, $P=\varepsilon_0\sum_n\chi^{(n)}E^n$ . Each nonlinear term is a product of fields, so it drives polarization — and hence radiation — at new frequencies formed from sums and differences of the input frequencies. The coefficients shrink by one factor of the atomic field $E_{at}$ per order, so nonlinearity needs intense light; and inversion symmetry forces every even-order $\chi^{(2n)}$ to zero, leaving $\chi^{(3)}$ as the lowest nonlinearity in ordinary isotropic matter.

Where this lives today

Many devices that shift the colour of laser light can be understood through terms in this nonlinear-polarization expansion. The $\chi^{(2)}$ of engineered crystals — periodically poled lithium niobate above all — powers the green laser pointer (1064 nm doubled to 532 nm), the optical parametric oscillators that tune coherent light across the infrared for trace-gas sensing, and the spontaneous down-converters that are the standard source of entangled photon pairs. The $\chi^{(3)}$ that survives in glass drives the self-phase modulation broadening a femtosecond pulse into an octave-spanning supercontinuum, the four-wave mixing that shuttles data between fibre wavelength channels, and the coherent Raman microscopies (CARS, SRS) that image tissue by its own vibrations. At the extreme strong-field edge, as the driving field approaches the atomic field scale $E_{\rm at}$ , the perturbative description can break down and strong-field physics takes over: electrons can be ionized, accelerated, and driven back to emit high harmonics extending toward the soft-X-ray regime. Attosecond science lives just past where equation (2) stops converging — a direct continuation of the same story about a field rivalling the one inside the atom.

The physical picture

Picture one electron in the well that binds it to its atom. A gentle field slides it off centre, the restoring force pulls back in exact proportion, and the induced dipole is a clean copy of the drive — linear optics. Drive it harder and the electron reaches the parts of the well where the two walls are not mirror images: the potential is stiffer on one side than the other. Now the excursion is a distorted, non-sinusoidal wiggle — flattened where the wall is soft, pinched where it is hard. Fourier-analyse it and you find overtones the pure drive never contained: a component at twice the frequency, a static offset, a third harmonic. Those overtones are the harmonics and mixing terms of nonlinear optics, radiated by the accelerating charge.

Figure

Left: the response

P(E)

is a straight line at low field, bending away as

E

nears the atomic scale. Right: a pure

\omega

drive produces a distorted response carrying DC and

2\omega

The power-series expansion is a bookkeeping device for that distortion. Each factor of $E$ costs roughly $E/E_{at}$ , so $\chi^{(2)}E$ is smaller than $\chi^{(1)}$ by that ratio and $\chi^{(3)}E^2$ smaller still. Only when a laser field becomes an appreciable fraction of the atomic field do the higher terms earn their keep — which is why nonlinear optics waited for the laser and remains the optics of intense light. Symmetry then does a remarkable amount of the physics for free. If the medium is unchanged under inversion $\mathbf r\to-\mathbf r$ — true of every gas, liquid, and glass, none having any built-in handedness — then reversing $E$ must reverse $P$ . An even power such as $E^2$ cannot manage that (it stays positive when $E$ flips), so its coefficient is forced to zero: $\chi^{(2)}$ , and every even order, vanishes. This one argument explains why frequency doubling demands a special crystal while the Kerr effect appears in ordinary glass. The next figure makes the consequence vivid.

Figure

One input frequency

\omega

sprays into a comb DC/

\omega

2\omega

3\omega

. Inversion symmetry is a kill switch: in a centrosymmetric medium the even orders (DC, 2ω) vanish, leaving only ω and 3ω.

The core equations

The edifice starts from one admission: at large fields the response is no longer proportional, so we expand it as an analytic function of the field. Keeping the general tensor structure — crystal symmetry lives in the indices, repeated indices summed — gives the constitutive relation that names every susceptibility at once.

P_i=\varepsilon_0\left(\chi^{(1)}_{ij}E_j+\chi^{(2)}_{ijk}E_jE_k+\chi^{(3)}_{ijkl}E_jE_kE_l+\cdots\right) \qquad (1)

where: $P_i$ — induced polarization density (C/m²); $E_j$ — driving field (V/m); $\varepsilon_0$ — vacuum permittivity, $8.854\times10^{-12}\,\mathrm{F/m}$ ; $\chi^{(1)}_{ij}$ — linear susceptibility (dimensionless, rank 2); $\chi^{(2)}_{ijk}$ — second-order (m/V, rank 3); $\chi^{(3)}_{ijkl}$ — third-order (m²/V², rank 4). The rank climbs by one each order because each extra factor of the vector field needs a fresh index to contract against. How fast does the series fall off? Each new power of the field must be measured against the only field the atom knows — the Coulomb field $E_{at}$ that binds its own electron — so trading one order for the next costs a factor $E/E_{at}$ . That single ratio governs the whole hierarchy of nonlinear effects.

P^{(n)}\sim\varepsilon_0\chi^{(n)}E^n,\qquad \chi^{(n+1)}\sim\frac{\chi^{(n)}}{E_{at}},\qquad E_{at}=\frac{e}{4\pi\varepsilon_0 a_0^2}\approx 5.14\times10^{11}\,\mathrm{V/m} \qquad (2)

where: $E_{at}$ — the field a valence electron feels at the Bohr radius $a_0=0.529\,\text{Å}$ ; $e$ — elementary charge. Setting $\chi^{(1)}\sim 1$ gives $\chi^{(2)}\sim 1/E_{at}\approx 1.9\,\mathrm{pm/V}$ — the right ballpark for real crystals (Derivation B). For fields well below the atomic scale, higher-order terms are often strongly suppressed and the lowest non-negligible orders usually dominate; however, the convergence of the expansion is controlled by the material's response and nearby resonances, not by $E_{\rm at}$ alone. Now the payoff. Because the nonlinear terms are products of oscillating fields, trigonometry turns them into new frequencies. Squaring a single wave is the cleanest case, producing two things at once — a static polarization and an oscillation at twice the drive frequency.

E(t)=E_0\cos\omega t\ \Rightarrow\ E^2=\tfrac12E_0^2\left(1+\cos 2\omega t\right) \qquad (3)

where: $E_0$ — real field amplitude; $\omega$ — drive frequency. The constant $\tfrac12E_0^2$ drives a DC polarization — optical rectification — while $\cos 2\omega t$ is a dipole radiating the second harmonic at $2\omega$ ; together $P^{(2)}=\tfrac12\varepsilon_0\chi^{(2)}E_0^2(1+\cos 2\omega t)$ . Feed in two colours instead and the cross term $2E_1E_2\cos\omega_1 t\cos\omega_2 t=E_1E_2[\cos(\omega_1+\omega_2)t+\cos(\omega_1-\omega_2)t]$ drives sum- and difference-frequency mixing — the whole parametric family. The third-order term behaves differently, and it survives even where $\chi^{(2)}$ is forbidden. Cubing the drive returns a piece at the original frequency plus a third harmonic.

E^3=E_0^3\cos^3\omega t=\tfrac34 E_0^3\cos\omega t+\tfrac14 E_0^3\cos 3\omega t \qquad (4)

where: $E_0,\ \omega$ as above. The $\tfrac34\cos\omega t$ piece adds a field-dependent contribution at $\omega$ , shifting the effective index in proportion to intensity — the optical Kerr effect of the companion article. The $\tfrac14\cos 3\omega t$ piece radiates the third harmonic. Both are odd-order, so they persist in glass. Finally, the structural fact that organises the entire subject: the even orders can be switched off by symmetry alone, with no reference to the size of any field.

\chi^{(2)}_{ijk}=0\quad\text{whenever the medium is centrosymmetric}\ (\mathbf r\to-\mathbf r\ \text{invariant}) \qquad (5)

where: $\chi^{(2)}_{ijk}$ — the third-rank second-order susceptibility. The argument is one line (Derivation A): under inversion both $P$ and $E$ flip sign, so $P\propto E^2$ demands $-P\propto(-E)^2=+E^2$ , i.e. $P=-P=0$ . Every even order dies the same death; only non-centrosymmetric crystals — quartz, KDP, BBO, LiNbO₃, GaAs — keep their $\chi^{(2)}$ .

Derivation AWhy inversion symmetry annihilates every even order

This is the single most consequential structural fact in $\chi^{(2)}$ optics, and it follows from symmetry alone — no dynamics, no numbers, no estimate of any field. The goal: show that if a medium is centrosymmetric — invariant under spatial inversion $r\to-r$ — then every even-order electric-dipole susceptibility vanishes, leaving $\chi^{(3)}$ as the lowest nonlinearity in the bulk. The proof applies the inversion operation to the constitutive relation and demands the material response be unchanged.

Step 1. Write the $n$ th-order polarization, $P^{(n)}_i=\varepsilon_0\chi^{(n)}_{ij_1\cdots j_n}E_{j_1}\cdots E_{j_n}$ — one polarization component set by a product of $n$ field components contracted with a rank- $(n{+}1)$ material tensor. Why: the whole nonlinear response is a sum of such terms, and inversion acts on each independently, so we can test them one order at a time.

Step 2. Perform the inversion. Position reverses, and because both $P$ and $E$ are polar (odd) vectors, each flips sign: $P_i\to-P_i$ , every $E_{j}\to-E_{j}$ . Why: this is the content of "invert the experiment" — reflect the apparatus through the origin and every genuine vector reverses.

Step 3. Demand invariance of the material tensor. If the medium is centrosymmetric the operation maps it onto itself, so $\chi^{(n)}_{ij_1\cdots j_n}$ is unchanged — that is the definition of the symmetry. Why: the coefficients describe the material, not the field; a material that looks the same after inversion cannot have them altered.

Step 4. Read off the consistency condition. The left side picked up one sign flip (from $P$ ), the right side $n$ (one per field), so $-P^{(n)}_i=(-1)^n P^{(n)}_i$ , i.e. $P^{(n)}_i=(-1)^{n+1}P^{(n)}_i$ . For even $n$ this reads $P^{(n)}=-P^{(n)}$ , forcing $P^{(n)}=0$ and $\chi^{(n)}\equiv 0$ ; for odd $n$ it is the empty $P=P$ and the term survives. Why: this is why $\chi^{(2)}=0$ in glass but $\chi^{(3)}$ does not — and why doubling needs a crystal with no centre of symmetry. *As a check: a general polynomial response satisfying $P(-E)=-P(E)$ forces all even coefficients to vanish, and the $(-1)^{n+1}$ sign pattern is reproduced order by order.*

Go deeper · Derivation B · 1 minSizing the nonlinearity: χ⁽²⁾ from the atomic field

Derivation BSizing the nonlinearity: χ⁽²⁾ from the atomic field

Symmetry says whether $\chi^{(2)}$ can exist; this card estimates how big it is, using nothing but the scale of the field an electron already lives in. The goal is to show that a one-line argument — nonlinearity switches on when the drive becomes comparable to the internal atomic field $E_{at}$ — predicts $\chi^{(2)}$ to the right order of magnitude, and to check that honestly against a real crystal. The strategy: demand that the linear and second-order polarizations become comparable exactly when $E$ reaches $E_{at}$ , which fixes $\chi^{(2)}$ relative to the order-unity linear susceptibility.

Step 1. Anchor the scale. Nonlinearity appears when the electron is driven through a displacement comparable to atomic size $a_0$ , needing a field comparable to the Coulomb field holding it there, $E_{at}=e/(4\pi\varepsilon_0 a_0^2)\approx 5.14\times10^{11}\,\mathrm{V/m}$ . Why: the anharmonic shoulders of the potential sit at $\sim a_0$ , so $E_{at}$ is where the "spring" visibly stops being linear.

Step 2. Impose the crossover. The linear and quadratic terms should become comparable, $\chi^{(1)}E\sim\chi^{(2)}E^2$ , precisely at $E\sim E_{at}$ , which gives $\chi^{(2)}\sim\chi^{(1)}/E_{at}$ . Why: by construction the perturbation series is a re-expansion of one smooth response, so its terms must all be of similar size right where the response goes nonlinear — at $E_{at}$ .

Step 3. Put in numbers. With $\chi^{(1)}\sim 1$ (an order-unity susceptibility for a dense dielectric), $\chi^{(2)}\sim 1/(5.14\times10^{11}\,\mathrm{V/m})\approx 1.9\times10^{-12}\,\mathrm{m/V}=1.9\,\mathrm{pm/V}$ . Why: this is a genuine prediction with no fitting — it uses only $e$ , $\varepsilon_0$ , and $a_0$ .

Step 4. Check it, honestly. KDP, a standard doubling crystal, has $d_{36}\approx 0.39\text{–}0.44\,\mathrm{pm/V}$ , i.e. $\chi^{(2)}=2d\approx 0.8\,\mathrm{pm/V}$ . Our $1.9\,\mathrm{pm/V}$ is high by only a factor of about two — remarkable for an argument that could have erred by many decades and that ignored the tensor character, dispersion, and the crystal's true $\chi^{(1)}\approx 1.2$ . Why: the estimate's job is not three-digit accuracy but to explain why $\chi^{(2)}$ values are universally picometres-per-volt rather than nanometres or femtometres — and it nails that. *As a check: $E_{at}=5.14\times10^{11}\,\mathrm{V/m}$ , the estimate $\chi^{(2)}\approx 1.9\,\mathrm{pm/V}$ , and the ratio to KDP's measured $d_{36}$ is a factor of $\approx 2$ – $2.5$ .*

Limiting cases

Case	Result	Meaning
Weak field, $E\ll E_{at}$	$P\approx\varepsilon_0\chi^{(1)}E$	Ordinary linear optics; the index is intensity-independent and no new colours appear.
Centrosymmetric medium	$\chi^{(2)}=0,\ \ P\approx\varepsilon_0(\chi^{(1)}E+\chi^{(3)}E^3)$	Gases, liquids, glasses: no second-order effects; the lowest nonlinearity is $\chi^{(3)}$ (Kerr, third harmonic).
Non-centrosymmetric crystal	$P^{(2)}_i=\varepsilon_0\chi^{(2)}_{ijk}E_jE_k\neq 0$	KDP, BBO, LiNbO₃, GaAs support second-harmonic and three-wave parametric processes.
Strong field, $E\sim E_{at}$	series diverges	Perturbation theory fails; ionization and high-harmonic generation take over ( $E=E_{at}$ needs $I\sim 3.5\times10^{16}\,\mathrm{W/cm^2}$ ).

Common trap“χ⁽²⁾ = 0 in glass means glass has no optical nonlinearity”

The vanishing of $\chi^{(2)}$ is a statement about even orders only, and it is easy to over-read. Inversion symmetry kills $\chi^{(2)},\chi^{(4)},\dots$ — so bulk glass makes no second harmonic and no optical rectification — but leaves every odd order untouched. Glass keeps a perfectly good $\chi^{(3)}$ , the workhorse of fibre optics: third-harmonic generation, the intensity-dependent Kerr index behind self-phase modulation and self-focusing, four-wave mixing, stimulated Raman and Brillouin scattering. Femtosecond supercontinuum and solitons in silica fibre are pure $\chi^{(3)}$ phenomena. "No $\chi^{(2)}$ " means "no second-order effects from the bulk," not "no nonlinearity" — and even that ban lifts at a surface, where the symmetry breaks and second-harmonic light reappears as a background-free interface probe.

Worked examples

1 · How far from the atomic field is a real laser? (confidence builder)

A focused continuous beam delivers an intensity of $I=1\,\mathrm{GW/cm^2}=10^{13}\,\mathrm{W/m^2}$ in vacuum ( $n=1$ ). Find the peak electric field, and compare it with the atomic field $E_{at}$ to see where nonlinear optics actually operates.

Step 1. Relate field to intensity. For a plane wave $I=\tfrac12 n\varepsilon_0 c\,E_0^2$ , so $E_0=\sqrt{2I/(n\varepsilon_0 c)}$ . The standard time-averaged Poynting relation; with $n=1$ it isolates the field an intensity carries.

Step 2. Substitute: $E_0=\sqrt{2(10^{13})/[(8.854\times10^{-12})(2.998\times10^{8})]}=\mathbf{8.68\times10^{7}\,V/m}$ — a gigawatt per square centimetre, enough to damage many materials, is tens of megavolts per metre.

Step 3. Compare with the atomic scale: $E_0/E_{at}=8.68\times10^{7}/5.14\times10^{11}=\mathbf{1.7\times10^{-4}}$ — under two ten-thousandths of the binding field. On the response curve of the physical-picture figure this sits essentially at the origin, deep in the linear regime.

Step 4. Turn it around: reaching $E=E_{\rm at}$ would need $I=\frac12\epsilon_0cE_{\rm at}^2\approx3.5\times10^{16}\ {\rm W/cm^2}$ , far above the damage thresholds of many ordinary optical materials. A 1 GW/cm² beam reaches $8.68\times10^{7}\,\mathrm{V/m}$ , only $1.7\times10^{-4}$ of $E_{at}$ — which is why the power series works: nonlinear optics is perturbation theory in a very small parameter.

2 · The size of the second-order correction (exam level)

For the same $1\,\mathrm{GW/cm^2}$ beam in a non-centrosymmetric crystal, estimate the ratio of the second-order to the linear polarization, $P^{(2)}/P^{(1)}$ , and comment on how such a tiny correction can still build a bright doubled beam.

Step 1. Form the ratio directly from the expansion: $P^{(2)}/P^{(1)}=\chi^{(2)}E_0^2/(\chi^{(1)}E_0)=(\chi^{(2)}/\chi^{(1)})E_0$ . Every extra order carries one more factor of $E$ , so the ratio is linear in the field.

Step 2. Using the atomic-scale estimate $\chi^{(2)}/\chi^{(1)}\sim1/E_{\rm at}$ from Derivation B gives $P^{(2)}/P^{(1)}\sim E_0/E_{\rm at}=1.7\times10^{-4}$ — the same dimensionless field ratio as in Example 1. This illustrates the expected smallness of the nonlinear correction for this estimate, although actual materials can differ substantially.

Step 3. Reconcile the smallness with reality. A per-pass correction of $10^{-4}$ sounds hopeless, but the doubled wave is coherently seeded at every plane, and when phase-matched those contributions add in amplitude — the amplitude grows linearly with length, the power as its square. The bridge to the companion article on second-harmonic generation.

Step 4. The moral: nonlinear optics is not about large responses but about small responses added in step. At $1\,\mathrm{GW/cm^2}$ , $P^{(2)}/P^{(1)}\approx 1.7\times10^{-4}$ — small per atom, but coherent accumulation over a phase-matched crystal turns it into efficient frequency conversion.

Physics of the portrait

The expansion above looks inevitable in hindsight, but someone had to build the systematic theory of how these waves feed one another inside a crystal — and to win a Nobel Prize for the spectroscopy the framework made possible.

Origins · 1 minNicolaas Bloembergen and how this was discovered

Nicolaas Bloembergen

1920–2017 · Harvard University (nonlinear optics)

Nicolaas Bloembergen was a Dutch-American physicist who first made his name elsewhere: as a graduate student he joined Purcell's group at Harvard weeks after the discovery of nuclear magnetic resonance, and the 1948 "BPP" paper he co-authored on nuclear relaxation remains a foundation of NMR. When the laser arrived he turned to the optics of intense light. With his Harvard colleagues J. A. Armstrong, J. Ducuing and P. S. Pershan he published in 1962 Interactions between Light Waves in a Nonlinear Dielectric, which laid out the coupled-wave theory of frequency mixing — the phase-matching bookkeeping, the Manley–Rowe photon-flux relations, and, presciently, the idea of periodically reversing the nonlinear coefficient to compensate a phase mismatch: the seed of modern quasi-phase matching. His 1965 monograph Nonlinear Optics named and organised the subject. Bloembergen shared the 1981 Nobel Prize in Physics with Arthur Schawlow (the other half went to Kai Siegbahn) for laser spectroscopy. He insisted that nonlinear optics be built on careful electromagnetic boundary conditions rather than hand-waving, remained active into his nineties, and died in 2017 at ninety-seven.

History

1960

— Maiman operates the first laser (ruby), providing the intense coherent fields nonlinear optics needs

1961

— Franken, Hill, Peters and Weinreich observe optical second-harmonic generation, the first nonlinear-optical experiment

1961

— Kaiser and Garrett observe two-photon absorption in CaF₂:Eu²⁺, a distinct third-order nonlinearity

1962

— Armstrong, Bloembergen, Ducuing and Pershan publish the coupled-wave theory of nonlinear dielectrics

1965

— Bloembergen's monograph Nonlinear Optics names and systematises the field

1981

— Bloembergen shares the Nobel Prize in Physics with Schawlow for laser spectroscopy

Going further

Texts. Boyd, Nonlinear Optics, 3rd ed., ch. 1 (the constitutive expansion and symmetry) · Saleh & Teich, Fundamentals of Photonics, ch. 21 (nonlinear optics) · Loudon, The Quantum Theory of Light, 3rd ed. · Hecht, Optics, 5th ed., §13.5 · Demtröder, Laser Spectroscopy, Vol. 1. Key numerical anchors: the atomic field $E_{at}=e/(4\pi\varepsilon_0 a_0^2)\approx 5.14\times10^{11}\,\mathrm{V/m}$ with $a_0=0.529\,\text{Å}$ ; the trigonometric identities $\cos^2\theta=\tfrac12(1+\cos2\theta)$ and $\cos^3\theta=\tfrac34\cos\theta+\tfrac14\cos3\theta$ underlie the frequency-mixing results; the scaling $\chi^{(n+1)}\sim\chi^{(n)}/E_{at}$ governs the order hierarchy; and the $\chi^{(2)}\approx 1.9\,\mathrm{pm/V}$ estimate compares with KDP's $d_{36}\approx 0.39$ – $0.44\,\mathrm{pm/V}$ .

Prerequisites. Electric Polarization and the Displacement Field · Dielectric Susceptibility and Permittivity · The Lorentz Oscillator Model · Maxwell's Equations in Dielectric Media. Leads to. The Refractive Index · Sum- and Difference-Frequency Generation · Phase Matching in Nonlinear Crystals.

If χ⁽²⁾ is only about 1 pm/V, how can a frequency doubler ever reach tens of percent efficiency?

Because the conversion is coherent, not a single scattering event. The $2\omega$ polarization is seeded in phase at every plane of the crystal, and when the crystal is phase-matched those seeds radiate in step: the harmonic amplitude grows linearly with distance, so its power grows as the square of the length. A per-atom response of $10^{-4}$ , summed over millimetres of matched crystal and boosted by high circulating intensity inside a cavity, routinely transfers most of the beam. The smallness of $\chi^{(2)}$ sets the required length and intensity, not the efficiency ceiling.

Is the power series in E actually convergent, or is it an asymptotic expansion?

Strictly it is asymptotic, and the dividing line is physical. For $E\ll E_{at}$ the terms fall off like powers of $E/E_{at}$ and a few of them describe the response to spectroscopic accuracy, so in practice the series is superbly useful. As $E$ climbs toward $E_{at}$ the higher terms stop shrinking, the perturbative picture fails, and one crosses into ionization and high-harmonic generation — a plateau of comparable harmonics, the opposite of a converging series. Read the expansion as "valid until the field rivals the one holding the electron in place," which for ordinary lasers is essentially always.

Why does inversion symmetry forbid even orders but spare the odd ones?

Track the signs. Under $\mathbf r\to-\mathbf r$ the polarization and the field both reverse, because both are polar vectors. A term $P\propto E^n$ then picks up one sign flip on the left (from $P$ ) and $n$ on the right (one per field), giving $-P=(-1)^n P$ . For even $n$ this says $P=-P$ , which only $P=0$ can satisfy, so the coefficient must vanish; for odd $n$ it says $P=P$ , an empty condition the term happily survives. The asymmetry between even and odd is thus not about magnitudes at all — it is a parity mismatch between a single odd $P$ and an even number of odd fields.

If bulk glass has no χ⁽²⁾, why do people report second-harmonic light from glass surfaces?

Because a surface is where the inversion symmetry that killed $\chi^{(2)}$ is broken. In the bulk every direction has its inverted partner and the even-order response averages to zero; but the topmost atomic layers see a different environment above and below, so locally there is no centre of symmetry and $\chi^{(2)}$ is nonzero. The signal therefore comes only from a nanometre-thick interface sheet — which is what makes surface SHG so valuable: intrinsically background-free and sensitive to adsorbates, molecular orientation, and buried interfaces that bulk probes cannot isolate.

How does this connect to the Lorentz oscillator model I already know?

Directly: the nonlinear susceptibilities are its anharmonic generalisation. The Lorentz model treats the bound electron as a damped harmonic oscillator, $\ddot x+\gamma\dot x+\omega_0^2 x=-(e/m)E$ , whose steady state gives the linear $\chi^{(1)}$ and all of ordinary dispersion. Add a small anharmonic term $a x^2$ to the restoring force and expand order by order: the square of the linear solution feeds back a source at $2\omega$ , whose amplitude is $\chi^{(2)}$ . The coefficient $a$ is nonzero only in an asymmetric potential — the microscopic statement of "non-centrosymmetric." So the same oscillator that explains the refractive index, given one asymmetric term, explains second-harmonic generation.

Why is E_at ≈ 5×10¹¹ V/m the right yardstick, rather than the field needed to ionize the atom?

Because $E_{at}=e/(4\pi\varepsilon_0 a_0^2)$ is the Coulomb field a valence electron already sits in at the Bohr radius — the field that shapes the binding potential, anharmonic shoulders and all. Nonlinear response turns on when the external field becomes an appreciable fraction of that internal field, because only then is the electron pushed into the asymmetric part of its well. The ionization field is numerically similar (same atomic scales), but $E_{at}$ is the correct yardstick for susceptibilities: it is the field against which each factor of $E$ is measured, which is why $\chi^{(n+1)}/\chi^{(n)}\sim 1/E_{at}$ .