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Suppose a homogeneous bulk material is unchanged when all spatial coordinates are reversed. Both electric polarization and electric field reverse under this inversion, so a local electric dipole response must obey $P(-E)=-P(E)$. Substituting the nonlinear optical susceptibility expansion givesFor every sufficiently small $E$, all even coefficients vanish. In the full tensor relation, two reversed input electric fields leave their product unchanged while the output electric polarization changes sign, forcing every bulk electric-dipole component of ${\chi }^{(2)}$ to vanish.

The qualification matters: the argument assumes inversion symmetry, a local bulk electric dipole response, and no applied bias that breaks that symmetry. A surface breaks bulk inversion symmetry; spatially varying or magnetic responses also lie outside this local electric-dipole model. Thus detecting second-harmonic generation at a surface or through a weaker magnetic contribution does not contradict the bulk selection rule. Conversely, absence of inversion symmetry permits some quadratic components but does not guarantee a nonzero component in every beam geometry.

Comb self-referencing compares two parts of one optical frequency comb. Coherent broadening supplies an octave, a factor of two in frequency. Second-harmonic generation doubles a red tooth; a photodiode detects its beat with an original blue tooth.^[ref] Applying the tooth formula givesThe doubled branch carries $2f_0$, while the original high-frequency branch carries $f_0$: their difference retains exactly one offset.

The displayed difference uses the order doubled tooth minus original tooth. An ordinary electronic frequency counter returns a nonnegative beat magnitude; spectral selection and known electronic shifts must identify whether it represents $f_0$, $f_r-f_0$, or a shifted version. Coherent broadening, overlapping photon polarization, spatial alignment and temporal overlap are required. Merely producing a broad patch of colored light does not establish a usable offset signal.

Common attenuation in third-order pulse correlation multiplies the entire input intensity profile by one constant factor before the correlation is formed. For the ideal signalreplacing $I$ by $kI$ gives $A_{3,k}(\tau )=k^3{A}_3(\tau )$. All unnormalized lobes scale together, so division by the central peak removes this common factor. The primary correlator paper defines the three-intensity-factor integral and implements it through second-harmonic generation and sum-frequency generation.^[ref]

The factor $k$ is an intensity transmission. An optical field-amplitude factor $a$ instead gives intensity factor $|a{|}^2$ and correlation factor $|a{|}^6$. Confusing these conventions changes the predicted power law. A spectral filter that reshapes or chirps the pulse is also not necessarily a common scalar attenuation.

An original example takes an ideal central signal 100 and a side lobe 1 in consistent detector-signal units. Halving the full input intensity gives central 12.5 and side 0.125; their ratio remains 0.01. In contrast, attenuation scaling of a third-order correlation ghost changes only the delayed replica and predicts different real-lobe and ghost powers. A common attenuator cannot perform that selective test.

A fixed background changes what normalization appears to show. Add one unremoved signal unit to both locations. Before attenuation the observed side/center ratio is $2/101\approx 0.0198$. After common intensity transmission $k=1/2$, it is $1.125/13.5=0.0833$. The apparent side feature has grown relative to the peak even though the optical profile did not change. Detector baseline subtraction and a verified linear readout are therefore part of the experiment.

Shot-to-shot intensity variation supplies another consequence. Let the common intensity multiplier be $K=1+\eta$, where $\eta$ has zero mean. ThenFor equally likely multipliers 0.9 and 1.1, the mean intensity is unchanged but the mean cubic signal is $1.03$ times the unit-intensity signal. This cubic intensity-noise bias in pulse correlation shows why cubing a mean input is not equivalent to averaging third-order signals. The calculation assumes the pulse shape is unchanged on each shot.

A controlled check records pulse intensity alongside the correlator signal, varies a broadband common attenuator, and tests cubic scaling in an unsaturated interval after background correction. Failure of that test can indicate detector nonlinearity, pulse reshaping or an uncontrolled background; it does not uniquely identify a particular fault. The ideal law applies only while the nonlinear gate and optical overlap retain their stipulated response.

Cubic intensity-noise bias in pulse correlation is the difference between the average of a third-order signal and the signal predicted from the average intensity when a fluctuating common intensity multiplier is cubed. Assume every shot has the same pulse shape, $I_j(t)=K_j{I}_0(t)$, with nonnegative scalar $K_j$. The ideal third-order integral givesThe three intensity factors follow from the second-harmonic generation and sum-frequency generation gate.^[ref]

Let $K=\mu +\delta$, with $\mathbb{E}[\delta ]=0$, variance ${\sigma }^2$ and third central moment $m_3$. Expanding the cube givesFor $\mu >0$, write the coefficient of variation as $v=\sigma /\mu$ and, when $\sigma >0$, skewness as $\gamma =m_3/{\sigma }^3$. The relative multiplier is $1+3v^2+\gamma v^3$. For zero variance the multiplier is one without needing to define skewness.

Equally likely $K=0.9$ and $1.1$ have mean one and give mean cubic multiplier $1.03$. Rare bright shots can matter much more: stipulated probabilities 0.9 and 0.1 at $K=0.8$ and $2.8$ also give mean one, butThe extra signal need not mean that the mean input intensity increased. These are original probability examples, not measurements reported in the correlator paper.

For unchanged pulse shape, common fluctuations multiply every delay by the same expected factor. Normalizing the averaged trace by its peak cancels that factor. Shape changes correlated with brightness destroy this simplification. For example, two equally frequent shot classes with central signals 1 and 8 and side/center ratios 0.01 and 0.10 give ratio of averaged signals $(0.01+0.8)/(1+8)=0.09$, whereas averaging the two ratios gives 0.055. They answer different questions.

A controlled measurement records a synchronized input-intensity monitor and detector background alongside each trace. Compare intensity-binned signals in an unsaturated range and test whether normalized shapes agree. Common attenuation in third-order pulse correlation predicts the cubic scaling only while the nonlinear gate, pulse shape and readout retain the assumed response. A mean power reading alone does not determine the required third moment.

An experimental optical physicist who coauthored a 1961 observation of optical second-harmonic generation. Passing intense laser light through quartz generated radiation with half the incident wavelength. Spectral separation of that weak new component supplied a direct test of the optical medium's nonlinear polarization.^[ref]

In 1961, Peter Alden Franken, Alan Hill, C. Wilbur Peters and Gabriel Weinreich focused a pulsed ruby laser through a red optical filter into quartz. The filter suppressed xenon-flash background. A quartz-prism spectrograph used red-insensitive Eastman Type 103 plates. A single shot recorded a feature near $347.2\mathrm{nm}$ from $694.3\mathrm{nm}$ input. Removing the quartz or substituting glass removed the feature; changing polarization of waves and orientation followed their predicted pattern. These controls supported second-harmonic generation; photographic density gave an estimated yield.^[ref]

A spectrograph turns output frequency into a recording position. Its spectral separation makes a faint new component distinguishable from bright input light. The reported rounded wavelengths do not constitute an independent precision measurement of the ratio two. Nor do these observations demonstrate perfect phase matching or complete conversion. The experiment and Nicolaas Bloembergen's later theoretical account are distinct contributions to nonlinear optics.

An experimental optical physicist who coauthored the observation of optical second-harmonic generation. A strong ruby-laser pulse passing through quartz produced light at twice the incident frequency. The experiment showed that a material's nonlinear polarization could mix an optical field with itself, turning a nonlinear response into a new observable color.^[ref]

Gaussian instrument broadening in third-order pulse correlation is the widening produced when an ideal third-order intensity autocorrelation is convolved with an additional Gaussian instrument response. It is a model of the measurement, not a universal correction for every correlator. The finite width of the optical nonlinear gate is already part of the ideal third-order trace and must not be counted again as an independent response.

Let a Gaussian pulse have intensity standard deviation ${\sigma }_I$. Its ideal trace is a Gaussian $C(\tau )=C_0\exp [-{\tau }^2/(2{\sigma }_3^2)]$ with ${\sigma }_3^2=3{\sigma }_I^2/2$. Take the additional normalized impulse response to beThe convolution $M=C\ast R$ isThe response preserves integrated trace area while reducing peak height and increasing variance. Since every Gaussian full width at half maximum has the same conversion factor from standard deviation,

For an original example, ${\tau }_I=80\mathrm{fs}$ and ${\tau }_R=60\mathrm{fs}$ give an ideal correlation width about $98.0\mathrm{fs}$ and a measured width about $114.9\mathrm{fs}$. The peak falls to approximately 0.853 of the ideal peak under this unit-area response. Dividing the measured width only by $\sqrt{3/2}$ would overestimate the pulse duration.

The correct Gaussian inversion is ${\tau }_I=\sqrt{(2/3)({\tau }_M^2-{\tau }_R^2)}$ when the radicand is positive. Its conditioning worsens near the response floor: the derivatives with respect to ${\tau }_M$ and ${\tau }_R$ are $(2/3){\tau }_M/{\tau }_I$ and $-(2/3){\tau }_R/{\tau }_I$. Small width errors can therefore cause large errors in a narrowly resolved pulse. Gaussian pulse-width deconvolution uncertainty must include uncertainty of the characterized response, with covariance if relevant.

The primary correlator used second-harmonic generation, sum-frequency generation, crossed beams and spatial imaging to encode delay. It did not assert that every additional broadening mechanism was a Gaussian convolution.^[ref] To apply this model, characterize the response under the same optical alignment, spectral acceptance and detector conditions. Non-Gaussian wings, clipping or unresolved side lobes require a different model; a smaller sampling step cannot by itself remove optical broadening.

Choose one driving electric field component and one responding electric polarization component. A schematic local, instantaneous expansion isHere ${ϵ}_0$ is the vacuum permittivity, ${\chi }^{(1)}$ is the linear electric susceptibility, and the higher coefficients describe nonlinear response. In SI, $P$ has units $\mathrm{C}{\mathrm{m}}^{-2}$, $E$ has units $\mathrm{V}{\mathrm{m}}^{-1}$, and ${\chi }^{(2)}$ has units $\mathrm{m}{\mathrm{V}}^{-1}$. The selected scalar coefficient depends on polarization of waves and crystal orientation; it is not a universal material number independent of geometry.

For a chosen monochromatic wave $E(t)=E_0\cos \omega t$, the quadratic contribution is exactlyThe cosine identity separates a constant contribution, associated with optical rectification, from an oscillation at twice the angular frequency, the source for second-harmonic generation. A term oscillating at $2\omega$ is a local driving source; efficient outgoing light additionally requires propagation and phase matching.

For a material with optical dispersion, the scalar instantaneous expansion is only a teaching model. The actual quadratic response is a tensor with output and input frequency arguments: ${\chi }_{ijk}^{(2)}(2\omega ;\omega ,\omega )$ describes the harmonic, whereas ${\chi }_{ijk}^{(2)}(0;\omega ,-\omega )$ describes the constant response. Their coefficients need not coincide. The cosine calculation establishes which frequencies are available, without assigning measured coefficients to any crystal.

Nonlinear optics studies the response of matter to light when the induced electric polarization is not proportional to the driving electric field. An intense laser can make these additional terms observable. The striking consequence is that a material can emit light at a frequency absent from the incident beam: second-harmonic generation doubles it. This extends a s usable optical spectrum without requiring a second lasing transition.

Franken and colleagues frequency-doubling experiment established an optical harmonic in 1961. Nicolaas S theoretical work developed the description of interacting light waves in nonlinear matter. His 1981 lecture connects a nonlinear constitutive relation with Maxwell's equations and explains compensation of optical dispersion by birefringence.^[ref]

Nonlinear polarization is the part of an induced electric polarization nonlinear in the driving electric field. A quadratic response multiplies field components, creating sums and differences of their frequencies; second-harmonic generation is the equal-frequency sum case. Conversion strength also depends on phase matching and the material response.

Optical pulse temporal contrast compares a pulse's peak with weaker light at another time. For a radiant power trace with peak $P_{\max }>0$, explicitly choose either the peak-to-local ratio $C(t)=P_{\max }/P(t)$ or its inverse $r(t)=P(t)/P_{\max }$. The first is at least one where the local power is positive; the second lies between zero and one. With a fixed transverse beam shape, the same ratio can be formed from light intensity. If the beam shape changes with time, specify the spatial sampling instead of identifying local intensity and total power.

A plateau at $10^{-3}$ of the peak has $r=10^{-3}$ and $C=1000$. Expressing this power ratio as optical contrast in decibels givesFor an amplitude ratio $A$ whose power ratio is $A^2$, the same definition gives $10{\log }_{10}(A^2)=20{\log }_{10}A$. Contrast also needs a delay or time range: an optical prepulse, an optical postpulse and a long optical pulse pedestal can have the same relative height but different positions and laser pulse energies. The pedestal energy fraction additionally requires integration over the contribution's duration.

A useful nonlinear measurement is third-order intensity autocorrelation. In an ideal instantaneous model, an input light intensity $I(t)$ generates a second-harmonic gate proportional to $I^2(t)$. Sum-frequency generation with a delayed fundamental replica gives a recorded signal proportional toThe instrument measures this cross-correlation, rather than simply reading $I(\tau )$. Finite gate width, phase matching, spatial overlap and detector response enter the interpretation.

The1998/99 instrument of J. Collier and colleagues used second-harmonic generation, a Wollaston prism and an optical biprism to form crossing fundamental and harmonic beams. A second crystal generated the sum-frequency signal, imaged through optical filters onto a charge-coupled device. Position encoded delay. An inserted glass etalon produced a delayed replica near $1.5\times 10^{-3}$ of the main signal, recorded on one side of the correlation peak. Figure4 combines window settings shifted with etalons.^[ref]

An independent ideal calculation explains both the asymmetry and a possible third-order pulse correlation ghost. Let $I(t)=g(t)+\epsilon g(t-d)$ represent two identical-shape pulses with well-separated supports and correlation lobes, with the second weaker and delayed. Squaring introduces no overlap term. If $A_g(\tau )=\int g^2(t)g(t+\tau )dt$, expansion givesThe delayed replica produces a lobe of relative order $\epsilon$, while the opposite-delay ghost has order ${\epsilon }^2$. For $\epsilon =0.01$, the scales are about $10^{-2}$ and $10^{-4}$ respectively. These values are predictions of the stated ideal model. Assigning every small correlation feature directly to a separate input pulse would miss this distinction.

A nondetection below the measurement floor establishes a bound under a detection model, rather than demonstrating that no light exists there. Finite dynamic range, detector baseline subtraction, delay coverage and resolution should accompany a reported contrast trace. The Sequoia manufacturer likewise distinguishes optical temporal resolution from the scan step; making that step smaller does not alone narrow the optical response.^[ref]

For bulk electric-dipole second-harmonic generation, the crystal must lack inversion symmetry and have a nonzero relevant component of its second-order nonlinear optical susceptibility. It should transmit both pump and harmonic, tolerate the required intensity, and allow phase matching by suitable birefringence or another phase-compensation mechanism. A large nonlinear coefficient, low optical loss and good optical quality are desirable. Noncentrosymmetry permits a response but does not guarantee one in every chosen geometry.

In the undepleted plane wave limit, contributions from successive layers give a harmonic amplitude proportional to ${\int }_0^L{e}^{i\Delta kz}dz$, where $\Delta k=k_{2\omega }-2k_{\omega }$. HenceTune orientation, temperature and pump polarization to maximize the nonlinear coupling and cancel phase mismatch. For continuous operation, an impedance-matched enhancement optical resonator, or intracavity doubling, raises the circulating intensity. Focus the Gaussian beam to balance intensity against diffraction and spatial walk-off; arbitrarily tight focusing shortens the useful overlap. Antireflection surfaces, stable resonance locking and efficient heat removal reduce avoidable loss and drift.

A crystal needs a nonzero effective second-order nonlinear optical susceptibility for the chosen polarizations. In the bulk electric-dipole approximation, centrosymmetry forbids this response. Transparency at the relevant frequencies and phase matching over the interaction length permit efficient buildup. Nonzero local generation can occur without exact phase matching, but contributions then tend to cancel as they propagate.

Write ${\psi }_j=k_{j}z-{\omega }_{j}t+{\varphi }_j$. Expanding the square and using the product-to-sum identity givesThese are respectively optical rectification, the two second-harmonic generation terms, sum-frequency generation and difference-frequency generation. Their phases and wavevectors are the corresponding sums or differences, not independently adjustable quantities.

For a vector field, $P_i^{(2)}=2{ϵ}_0{\sum }_{jk}{d}_{ijk}{E}_j{E}_k$. In the stated single-frequency amplitude convention, $E_j=(A_j{e}^{i\psi }+A_j^{\ast }{e}^{-i\psi })/2$. If $P_i^{(2\omega )}=[{\mathcal{P}}_i{e}^{2i\psi }+{\mathcal{P}}_i^{\ast }{e}^{-2i\psi }]/2$, comparison givesFor a real instantaneous tensor the zero-frequency contribution is $P_i^{(0)}={ϵ}_0{\sum }_{jk}{d}_{ijk}\mathrm{Re}(A_j{A}_k^{\ast })$. In a dispersive treatment, the tensors must carry their respective input and output frequency arguments.

The identical input-frequency pair is symmetric under $j\leftrightarrow k$, allowing contracted nonlinear susceptibility notation: replace the six pairs $xx,yy,zz,yz,zx,xy$ by $J=1,\dots ,6$. ThusThis contraction of a symmetric pair does not turn the physical susceptibility into a rank-two tensor. The factors of two account for both off-diagonal input orderings, and the prefactor follows from the question's explicit $1/2$ field convention.

For the proposed type-I process, energy conservation requires ${\omega }_1={\omega }_2+{\omega }_3$. HenceThe extraordinary pump and two ordinary output waves can be collinearly phase matched ifPut $n_{\ast }={\lambda }_{1}S$. Substituting the supplied angular extraordinary index and solving for ${\sin }^2\theta$ yieldsThe required index $n_{\ast }\simeq 2.2143$ lies between $2.1782$ and $2.2579$, so a real angle exists. Using the rounded 2922 nm wavelength in the supplied data gives $\theta \simeq 46.94^{\circ }$; retaining the energy-conserving 2922.25 nm value gives about $46.98^{\circ }$. Their common meaningful scale is about 47 degrees.

The spatial walk-off angle is the angle between the extraordinary wave's Poynting vector and its wavevector. For an optic-axis angle $\theta$, the extraordinary dispersion surface givesHere the indices are those of the 780 nm pump. Since this is a negative uniaxial crystal, its energy flow is farther from the optic axis than its wavevector. The ordinary waves' energy flows follow their wavevectors. The pump therefore drifts out of spatial overlap by roughly $L\tan \rho \simeq 0.0357L$, about $36\mu \mathrm{m}$ over a millimetre. This limits the useful nonlinear interaction length even when phase matching is exact.

In second-harmonic generation, the quadratic nonlinear polarization contains products of pairs of comb components. Their angular frequencies areincluding $2{\omega }_p$ when $p=q$. Original comb components have angular frequencies ${\omega }_r={\omega }_{\mathrm{ce}}+r\Delta \omega$, henceWhen the doubled low-frequency region overlaps the original high-frequency region, choosing $r=p+q$ gives the beat at $|{\omega }_{\mathrm{ce}}|$. More generally the nearest positive beat has angular frequencywhich is $|{\omega }_{\mathrm{ce}}|$ for the stipulated small offset. For $0\le {\omega }_{\mathrm{ce}}<\Delta \omega$ it is $\min ({\omega }_{\mathrm{ce}},\Delta \omega -{\omega }_{\mathrm{ce}})$, provided the corresponding teeth are in the overlap.

In comb self-referencing, overlap the doubled red part and the original blue part on a photodiode. The beat measures the carrier-envelope offset frequency. Feedback on an intracavity dispersion control or pump setting can set ${\omega }_{\mathrm{ce}}=0$ modulo $\Delta \omega$, giving $\boxed{{\varphi }_{\mathrm{slip}}=0(\mathrm{mod}2\pi )}$. A phase-sensitive lock or calibrated approach to zero resolves the sign ambiguity of an unsigned beat measurement.^[ref]

The coherent optical spectrum must span an octave, ${\omega }_{\max }\gtrsim 2{\omega }_{\min }$, with sufficient overlapping spectral strength after second-harmonic generation. This calls for very short broadband optical pulses, or coherent external spectral broadening. The two branches must preserve optical coherence, overlap in time and polarization, and have adequate phase matching; bandwidth alone does not guarantee a detectable beat.

Second-harmonic generation doubles optical frequency, hence halves the vacuum wavelength. It requires a nonzero effective second-order nonlinear optical susceptibility for the selected fields, overlap in the nonlinear medium, and enough intensity and interaction length for a useful signal. Bulk electric-dipole second-order response vanishes in an inversion-symmetric medium. Low absorption at both frequencies and phase matching prevent generated amplitudes from being lost or canceling during propagation. Pulses additionally need suitable temporal overlap and bandwidth acceptance.

For the stated lossless amplitude equations, choose real $\alpha$, input $A_0=|A_0|e^{i\varphi }$ and no second-harmonic seed. Perfect phase matching admitsThe second equation follows by substituting the first into $A_{\omega }^{\prime }=i\alpha A_{2\omega }{A}_{\omega }^{\ast }$. Its derivative then equals $i\alpha A_0^2{\mathrm{sech}}^2(\lambda z)$, verifying the other coupled equation. Equivalently, ${\lambda }^2={\alpha }^2|A_0{|}^2$; changing the sign of $\lambda$ does not change the pump profile. The intensity convention in the question therefore givesTheir sum is $I_0$, as follows directly from the supplied equal-coupling equations. This uses their normalization; physically normalized photon-flux equations can place frequency factors in the two coupling coefficients. A nonzero input harmonic would require a different boundary-value solution.

For an undepleted pump at nonzero mismatch, direct integration givesHere $\mathrm{sinc}x=\sin x/x$. The perfectly matched small-signal curve grows quadratically and its depleted continuation saturates. For $\Delta kL=\pi$, destructive phase accumulation reduces output; in the undepleted limit its endpoint is $4/{\pi }^2$ of the matched small-signal output. For $|\Delta k|L\gg \pi$, successive regions alternately generate and cancel harmonic amplitude, giving rapid low-amplitude oscillations with spatial period $2\pi /|\Delta k|$. For substantial depletion, the complete curve also depends on $\alpha \sqrt{I_0}L$; specifying mismatch alone does not determine a unique numerical sketch.

For the separated crystals, use the undepleted approximation and perfect matching inside each crystal. Up to a common output phase, the harmonic generated in the first crystal contributes $i\alpha A_0^{2}L{e}^{ik(2\omega )d}$, while the second crystal contributes $i\alpha A_0^{2}L{e}^{2ik(\omega )d}$. The different phases come from propagation of an already generated harmonic versus propagation of two pump factors through the gap. Ideal antireflection coatings remove the multiple-reflection complication. ThusThis is two-crystal second-harmonic interference. Maxima require $\Delta k_{g}d=2\pi m$. With nonzero dispersion, their spacing is $2\pi /|\Delta k_g|={\lambda }_0/[2|n(\omega )-n(2\omega )|]$, where ${\lambda }_0$ is the pump vacuum wavelength. If the gap has no index difference between these frequencies, every separation is constructive within this ideal propagation model.

Finite depletion makes the amplitude entering the second crystal smaller and permits coherent backconversion, so the simple cosine-squared envelope no longer applies and the total converted intensity cannot exceed $I_0$. The gap phase remains periodic. In the ideal identical lossless case the constructive setting is equivalent to a matched crystal of length $2L$, giving $I_0{\tanh }^2(2\alpha \sqrt{I_0}L)$. The opposite relative phase can make the second crystal reverse the first conversion, restoring the pump and leaving zero harmonic. Thus depletion does not necessarily fill an ideal destructive minimum; unequal crystals, loss or other phase errors can spoil that cancellation.

For a second-order response, $P_i^{(2)}({\omega }_3)={ϵ}_0{\sum }_{jk}{\chi }_{ijk}^{(2)}({\omega }_3;{\omega }_1,{\omega }_2)E_j({\omega }_1)E_k({\omega }_2)$, with the appropriate frequency conservation and negative-frequency convention for difference-frequency processes. Interchanging the two input field-frequency pairs gives intrinsic permutation symmetry of nonlinear susceptibility:For second-harmonic generation the input frequencies agree, so the last two Cartesian indices are symmetric and can be contracted into six input-pair labels. This reduces the unrestricted 27 Cartesian entries to at most 18 before crystal symmetries are imposed.

Crystal point-group transformations must leave the material tensor invariant and can force entries to vanish or become related. Under spatial inversion both electric field and polarization reverse, whereas the field product is even. A centrosymmetric bulk therefore has zero electric-dipole ${\chi }^{(2)}$. Surfaces and interfaces can break inversion symmetry, and higher-multipole processes are outside that simple prohibition.

Kleinman symmetry is a stronger approximate permutation symmetry used sufficiently far from resonances when dispersion can be neglected over the participating frequencies. It is not justified merely because two input frequencies are equal; resonant or appreciably dispersive media need the full frequency-dependent relations. These distinctions determine which polarization and propagation geometries have nonzero effective coupling, in addition to whether they satisfy phase matching.^[ref]

Peter Alden Franken coauthored the 1961 observation of optical second-harmonic generation. Intense ruby-laser light passing through quartz produced radiation at twice the incident frequency. A quadratic contribution to nonlinear polarization explains why this is possible: squaring a sinusoidal electric field produces a term at twice its frequency. The experiment made frequency conversion a measured consequence of nonlinear material response.^[ref]

A physicist.^[ref]

Phase matching makes wave amplitudes emitted at different depths add coherently. Consider collinear plane waves in a uniform transparent material of length $L$, with fixed polarization of waves and nonzero selected nonlinear optical susceptibility. Assume negligible pump depletion, negligible absorption, an initially absent harmonic, and a wide beam for which diffraction is negligible. The pump's wave amplitude is then constant throughout the interaction.

The pump electric field has spatial wave phase $k_{\omega }z$; its quadratic source has spatial wave phase $2k_{\omega }z$. The freely propagating harmonic has wave number $k_{2\omega }$. Define the second-harmonic phase mismatch byFor the chosen complex-field convention, the outgoing harmonic wave amplitude, up to an overall constant and propagation phase, is proportional toChoosing the conjugate complex-field convention reverses the exponent's sign without changing the wave intensity. Taking the squared complex modulus givesHere $C$ is fixed for the chosen modes and material; its dimensions accommodate the stated wave intensity convention. The quadratic pump dependence follows because the source wave amplitude is proportional to $E_0^2$, whereas $I_{\omega }$ is proportional to $E_0^2$. The sinc function comes from summing contributions along the whole material, just as a finite aperture produces diffraction through addition of complex amplitudes.

At $\Delta k=0$, the integral is $L$ and the harmonic wave intensity grows as $L^2$. At $\Delta kL=2\pi$, all those contributions cancel. This cancellation needs no absorption: their wave phases wind through a full turn. At high conversion, the constant-pump assumption fails, so the $L^2$ law cannot continue indefinitely. Tight focusing, finite optical pulses, absorption, and transverse separation of modes also require a more complete model.

Optical dispersion generally makes $n_{2\omega }\ne n_{\omega }$. In a suitable crystal, birefringence allows modes with different polarization of waves to experience different refractive indices, making their selected values equal. The appropriate crystal direction and allowed nonlinear optical susceptibility component must both work.^[ref]

For a worked calculation, choose $L_0=1\mathrm{mm}$ and $\Delta k=\pi \mathrm{m}{\mathrm{m}}^{-1}$. At $L=L_0$, the harmonic wave intensity is $4/{\pi }^2\simeq 0.4053$ of its perfectly matched value. At $L=2L_0$ it vanishes in this model. More material can therefore reduce the output. If instead $\Delta k=0$, doubling $L$ multiplies output by four. The two curves retain that common normalization, so their different peak heights are meaningful.

Second-harmonic generation converts an incident optical frequency $\nu$ into $2\nu$ through a material's nonlinear response. In the photon description of an ideal lossless conversion, two input photons supply one output photon:Here $h$ is the Planck constant. The relation states conservation of energy; the material mediates the interaction. It does not say that each input photon independently splits its wavelength in half.

For vacuum wavelengths, ${\lambda }_{0,\omega }=c/\nu$ and ${\lambda }_{0,2\omega }=c/(2\nu )$, where $c$ is the speed of light. HenceThe second relation uses the refractive index for each propagating mode and frequency. Within a medium with optical dispersion, the wavelength need not halve. For an invented example, ${\lambda }_{0,\omega }=800\mathrm{nm}$, $n_{\omega }=1.5$, and $n_{2\omega }=1.6$ give $400\mathrm{nm}$ in vacuum, but in the material the two wavelengths are $533.33\mathrm{nm}$ and $250\mathrm{nm}$; their ratio is $0.46875$.