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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.

A finite beam has an intensity gradient across its wavefront. In an instantaneous optical Kerr model, $n(r)=n_0+n_{2}I(r)$. A bright centre therefore sees a larger index for $n_2>0$, accumulating more phase than the wings and producing self-focusing. Negative $n_2$ gives self-defocusing. For a short medium and Gaussian $I(r)=I_0{e}^{-2r^2/w^2}$, expand the nonlinear phase near the axis:Compare with the in-medium thin-lens phase $-k{r}^2/(2f)$, where $k=n_0{k}_0$. The Kerr lens has $1/f=4n_2{I}_{0}L/(n_0{w}^2)$. This local quadratic fit describes the central lens, not the whole Gaussian phase or a guaranteed aberration-free focus.

Propagation combines diffraction with the nonlinear phase. Choose $|A{|}^2=I$ and a scalar paraxial envelope, givingThe diffraction length is of order $k{w}^2$, while the nonlinear length is $1/(|\gamma |I_0)$. Their competition depends on total beam power as well as spot size. In two transverse dimensions, balancing them gives a critical power scale $P_{\mathrm{cr}}\sim {\lambda }_0^2/(n_0{n}_2)$ for positive $n_2$; the numerical constant requires a specified transverse profile and critical solution. In the ideal unsaturated cubic model, sufficiently strong focusing can collapse rather than form a stable beam. Real saturation, absorption, ionization or nonparaxial propagation can stop that ideal singularity; the appropriate mechanism depends on the material and pulse regime.

A broad focusing beam can also break into filaments. Perturb the uniform solution $A_0{e}^{i\gamma I_{0}z}$ by real amplitude and phase components with transverse wave number $q$. Linearization gives the transverse modulational instability rateFor positive $\gamma$, modes with $0<q^2<4k\gamma I_0$ grow. diffraction suppresses very fine perturbations. A finite beam's boundaries and available propagation length limit which filaments develop. The defocusing sign has no growing band in this simple instantaneous local model.

A spatial optical soliton instead has a profile whose nonlinear focusing balances diffraction. A one-transverse-dimensional cubic focusing model supports a stationary bright solution; the existence of that solution does not establish robust two-dimensional cubic self-trapping. A beam can also act on another finite beam through cross-phase modulation, producing an induced lens or steering it towards a high-index region. Unequal polarization response can rotate or reshape the profile. Temporal self-phase modulation is a related consequence of a time-dependent intensity, but should be distinguished from these transverse effects.

For an experimental check, vary power at fixed input waist and compare near- and far-field profiles with a low-power diffraction reference. Repeat with a second beam blocked to separate cross-phase modulation from self-action. Vary pulse duration or repetition rate to test thermal lensing against a prompt electronic response, and monitor transmission to distinguish nonlinear refraction from preferential absorption. A smaller output spot alone does not establish a soliton or a pure Kerr response.