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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.

A crystal has centrosymmetry when its crystallographic point group contains spatial inversion. Its bulk electric-dipole polarization law must then satisfy $P(-E)=-P(E)$. Substituting a power expansion in E makes every even-order coefficient vanish, including the second-order nonlinear optical susceptibility. A surface, an external symmetry-breaking field or a response beyond the electric-dipole approximation can change the conclusion.

Difference-frequency generation produces a wave at the positive difference ${\omega }_i={\omega }_p-{\omega }_s$ of two incident angular frequencies, through a second-order nonlinear optical susceptibility. The product of two oscillating electric fields contains both sum and difference frequencies, and the corresponding polarization density drives the generated wave.

For plane waves in a uniform medium, coherent buildup is favored by ${\mathbf{k}}_p={\mathbf{k}}_s+{\mathbf{k}}_i$, the relevant phase matching condition. Two shorter-wavelength inputs can thereby generate longer-wavelength infrared radiation. Collecting the output after removing both incident frequencies distinguishes frequency generation from scattered pump light.^[ref]

A third-order nonlinear optical susceptibility produces a polarization term proportional to $E_1{E}_2{E}_3^{\ast }$. Its temporal and spatial phases give the displayed output frequency and the source wave vector ${\mathbf{k}}_1+{\mathbf{k}}_2-{\mathbf{k}}_3$. Efficient emission also requires that this source match a propagating optical mode. The pump waves supply the generated energy.

An intensity-dependent refractive index produced by the optical field itself. In a simple instantaneous isotropic model, $n=n_0+n_{2}I$. A transversely nonuniform intensity produces self-focusing for positive $n_2$ and self-defocusing for negative $n_2$. Temporal intensity variation produces self-phase modulation. These are consequences of a third-order nonlinear optical susceptibility, distinct from a static applied-field electro-optic Kerr effect.

A static or slowly varying electric polarization produced by nonlinear response to an oscillating optical electric field. The constant term in nonlinear optical susceptibility is its simplest model. An optical pulse has a varying envelope, so its cycle-averaged quadratic response can vary during the pulse rather than remaining constant forever. This term alone does not imply a radiated zero-frequency wave.

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.

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.

The polarization generated by the relevant third-order nonlinear optical susceptibility is proportional to $E_1{E}_2{E}_3^{\ast }$. Multiplication of its temporal and spatial wave phase factors givesThe two opposing pump arrows in the original diagram mean ${\mathbf{k}}_2=-{\mathbf{k}}_1$. Thus the mixing source points along $-{\mathbf{k}}_3$, and its conjugate signal amplitude drives a backward field. This is counterpropagating-pump phase matching in four-wave mixing.

There is a phase matching qualification to the initial frequency statement. In the stipulated nondispersive isotropic medium, $|{\mathbf{k}}_j|=n{\omega }_j/c$. Exact equality of the backward free-wave vector to $-{\mathbf{k}}_3$ therefore requires ${\omega }_4={\omega }_3$, hence ${\omega }_3={\omega }_P$. For unequal frequencies the backward direction remains the geometry suggested by the source, but the longitudinal phase mismatch has magnitude $2n|{\omega }_P-{\omega }_3|/c$. Efficient generation over a finite length requires that mismatch times $L$ be small; a sinc function factor suppresses conversion otherwise. Nondispersion does not remove the magnitude mismatch. The remainder of the calculation is explicitly degenerate four-wave mixing, so this difficulty disappears.

At the pump frequency, for the strong field $E_1$, self-phase modulation and cross-phase modulation by $E_2$ dominate the weak signal's influence. They change its propagation wave phase through its own intensity and the other pump's intensity. With continuous uniform plane waves, these are nonlinear wave phase shifts; finite pulses acquire temporal chirp, and finite transverse beams can acquire a lens. No universal factor of two should be assigned between the two shifts without specifying their polarization contraction and the field-amplitude convention.

Write $u={\tilde{A}}_3$ and $v={\tilde{A}}_4$. Differentiating the exponential substitutions cancels the terms proportional to $2\kappa$:Consequently $u^{\prime }=i\kappa v^{\ast }$ and $v^{\prime }=-i\kappa u^{\ast }$. Conjugate the latter before eliminating a field: $(v^{\ast }{)}^{\prime }=i\kappa u$, so $u^{\prime \prime }=-{\kappa }^{2}u$. The backward four-wave-mixing boundary problem requires opposite-end boundary conditions: $u(0)=a$, where $a=A_3(0)$, and $v(L)=0$. The transfer matrix isAt $L$, $0=ia\sin \kappa L+b\cos \kappa L$, so $b=-ia\tan \kappa L$. Substitution givesThe printed target expression for $A_4$ has the opposite overall sign. This is present in the original PDF, not just the transcription. With its stated first-order equations, the derivative at the far face must be $v^{\prime }(L)=-i\kappa a^{\ast }/\cos \kappa L$. The boxed result has that derivative; the printed expression has its negative. Reversing the convention for the mixing coefficient could reverse the generated phase, but would also change the coupled equations. The result above satisfies the equations and both boundary conditions together.

At the entrance, $A_4(0)=i{a}^{\ast }\tan \kappa L$, so the returned amplitude is proportional to the complex conjugate of the signal. An ordinary ideal mirror gives $ra$, with a fixed reflection coefficient $r$, rather than $a^{\ast }$. The mixing pumps supply both the extra energy and a wave phase reference. The ideal conjugate reflectivity and signal transmission areThe constancy of $|u{|}^2+|v{|}^2$ along the medium is another direct check. The relation between the incoming signal and outgoing waves allows net output energy to come from the pumps. As $\kappa L\to \pi /2$, the undepleted-pump approximation fails: the predicted fields would extract an appreciable pump fraction. This is the ideal four-wave-mixing oscillation threshold; saturation, pump depletion and noise seeding replace the divergent linearized solution. It is not an unlimited source of reflected power. Amplified phase-conjugated propagation and parametric oscillation are also the effects identified in the primary analysis by Amnon Yariv and David M. Pepper.^[ref]

For the aberrating linear medium, assume its permittivity is real and stationary at this frequency. Taking the complex conjugate of the forward-envelope equation givesThis is the backward-envelope equation with carrier $e^{-i{k}_{3}z}$. Launching the conjugate boundary field therefore permits retracing of the distorted wave. In a transverse lossless phase screen model, let transmission be $t(x)=e^{i\psi (x)}$. A return proportional to $(tU{)}^{\ast }$ acquires $t$ on its second passage and becomes $t(tU{)}^{\ast }=U^{\ast }$. Thus the aberration phase cancels. A mirror instead gives $r{t}^{2}U$, adding the aberration twice; merely reversing the central propagation direction does not reverse the wavefront phase.

The one-dimensional printed equation describes longitudinal variation; genuine transverse aberrations require the corresponding transverse Helmholtz equation or paraxial operator, for which the same real-coefficient conjugation argument applies. Exact reconstruction requires adequate aperture, the correct conjugate amplitude and polarization, uniform pump-product phase, and an unchanged reciprocal lossless medium. Absorption, missing scattered modes or a changed aberrator prevent a universal reconstruction. This is phase conjugation and thin phase-plate cancellation by phase conjugation under specified conditions, rather than reversal of every possible propagation loss.

Periodically poled lithium niobate is lithium niobate whose ferroelectric domain orientation is reversed periodically along an optical interaction path. Reversal changes the sign of the relevant second-order nonlinear optical susceptibility, supplying the spatial modulation needed for quasi-phase-matching.

For a uniform collinear conversion mismatch $\Delta k$, a first-order poling period $\Lambda$ can compensate it through $\Delta k-2\pi /\Lambda \simeq 0$, with signs chosen consistently. The generated contributions then reinforce over many domains instead of repeatedly canceling. The poling period, temperature and input wavelengths determine the useful conversion band; domain defects and finite length affect efficiency. ^[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.

In spontaneous parametric down-conversion, a strong pump drives a second-order nonlinear optical susceptibility, producing paired output photons in initially unseeded modes. The two outputs are conventionally called the signal photon and idler photon. They need not have equal frequencies. For a monochromatic pump, conservation of energy gives

Efficient bulk generation also requires phase matching: the nonlinear source and emitted waves must remain in phase. In the ideal plane-wave geometry this is ${\mathbf{k}}_p={\mathbf{k}}_s+{\mathbf{k}}_i$. Different refractive indices and propagation directions determine which pairs satisfy it. A finite interaction length supplies a mismatch bandwidth rather than an infinitely sharp wavevector constraint. ^[ref]

At weak gain, the SPDC pair state contains a large vacuum component and a small pair amplitude. Detecting one output conditions the other, producing a heralded single-photon source. This is probabilistic: stronger pumping also increases unwanted multiple-pair events. Experiments collect two output modes with optical filters and single-photon detectors, comparing coincident detections with background counts. ^[ref]

An interaction among three waves mediated by a second-order nonlinear optical susceptibility. In sum-frequency generation, two input angular frequencies satisfy ${\omega }_3={\omega }_1+{\omega }_2$. The same coupling permits reverse conversion. Phase matching determines whether successive generated contributions reinforce; the Manley-Rowe relations express the associated lossless exchange constraints.