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Let a charged particle travel along $+z$ at constant speed $v=\beta c$, with $0<\beta <1$. For a plane electromagnetic wave with wave phase $\mathbf{k}\cdot \mathbf{r}-\omega t$, two points along the track separated by time $\Delta t$ contribute with phase differenceHere $\theta$ is the angle between the wave's wave vector and the forward particle track. Constructive addition along a long uniform track requires $kv\cos \theta =\omega$. ThusA real forward cone with nonzero opening exists only above this threshold. At $\beta n=1$ the cone closes and the ideal Frank-Tamm formula gives zero yield. For $\beta n<1$ there is no allowed angle, rather than an imaginary observable cone. The requirement is phase matching; substituting the group refractive index would describe a different condition.

There is a second geometric derivation when $n$ is constant. In time $\Delta t$, the charge advances $OP=v\Delta t$, while an earlier spherical phase front has radius $OQ=c\Delta t/n$. Its tangent envelope passes through the current position $P$. The radius $OQ$ is normal to the envelope, so the right triangle gives $\cos \theta =OQ/OP$. Rotating about the track makes a cone. Its surface half-angle $\eta$ is complementary to the emission angle $\theta$, not equal to it.

In a dispersive medium, apply the phase matching derivation separately at each angular frequency. The circle construction uses one fixed phase speed; it does not determine the envelope of a broadband wave packet. In particular, group velocity controls packet propagation under its usual narrow-band assumptions, and the simple complementary-angle construction need not describe a dispersive pulse front. A finite track also broadens the ideal directional selection: adding contributions over duration $T$ gives a factor proportional to $\sin (\Delta T/2)/\Delta$, with $\Delta =kv\cos \theta -\omega$, rather than an infinitely sharp direction.

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]

For incident SV angle $\theta$, tangential phase matching gives $\sin \varphi =(c_P/c_S)\sin \theta$ for the reflected P-wave. It is evanescent when $\sin \theta >c_S/c_P$.

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.

G. Jonušauskas coauthored the 1992 beta barium borate experiment with A. Dubietis and A. Piskarskas, demonstrating parametric amplification of chirped and stretched femtosecond pulses.^[ref]

An optical parametric amplifier must satisfy both frequency relations and phase matching. For a pump, signal and idler with angular frequencies ${\omega }_p,{\omega }_s,{\omega }_i$, the three-wave process obeys ${\omega }_p={\omega }_s+{\omega }_i$. Coherent buildup also depends on their wavevectors: ideally ${\mathbf{k}}_p={\mathbf{k}}_s+{\mathbf{k}}_i$. Matching frequencies alone does not ensure that fields generated at different positions add constructively.

For an original undepleted-pump, weak-conversion illustration, suppose the driving amplitude is uniform along a crystal of length $L$. A scalar phase mismatch $\Delta k$ makes the generated field proportional toAt exact phase matching the limiting amplitude is proportional to $L$. At $\Delta kL=2\pi$ the idealized integral is zero because contributions cancel. This elementary coherent-sum model explains the need to control crystal orientation and optical frequencies; it is not the full high-gain solution or a reported setting from Jonušauskas's experiment.

For a downward transverse wave in an isotropic plasma, $E_z/E_x=\tan {\theta }_t$ fixes the transmitted direction. Tangential phase matching gives $\sin {\theta }_i=n_p\sin {\theta }_t$. Continuity of tangential electric and magnetic fields then separates incident and reflected amplitudes. A measured transmitted electric amplitude alone cannot be identified with the incident one; mean power for peak phasors includes a factor of one half, and interface-normal flux includes a cosine.

N. Boeuf and collaborators developed calculations of non-collinear phase matching in uniaxial and biaxial nonlinear crystals. The NIST-hosted 1999 draft extended calculations beyond cases where interacting waves all propagate along one line.^[ref]

For spontaneous parametric down-conversion, the pump, signal and idler satisfy ${\omega }_p={\omega }_s+{\omega }_i$ and ${\mathbf{k}}_p={\mathbf{k}}_s+{\mathbf{k}}_i$. The second condition is a vector relation, so crystal anisotropy and propagation angles jointly constrain possible outputs. Solving only the frequency equation does not establish an efficient optical interaction.

A crystal used for its nonlinear optics response, allowing processes such as spontaneous parametric down-conversion when the appropriate phase matching is satisfied.

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 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.

An optical parametric amplifier transfers energy from a pump beam into a signal and an idler through nonlinear optics. For the common three-wave mixing process, their angular frequencies obey ${\omega }_p={\omega }_s+{\omega }_i$. A strong pump and a weaker signal enter a nonlinear medium; the signal grows and an idler is generated.

Efficient growth requires phase matching so that contributions generated along the medium reinforce. Changing crystal orientation or temperature can tune the matched frequencies. Measuring the signal with and without the pump tests amplification, while observing the corresponding idler checks the frequency-conversion mechanism. The pump supplies the extra energy; the process does not require a stored population inversion.^[ref]

Optical parametric chirped pulse amplification combines the stretch–amplify–compress sequence of chirped pulse amplification with an optical parametric amplifier. A stretched signal pulse overlaps a strong pump pulse in a nonlinear crystal. Three-wave mixing transfers pump energy into the signal and an idler; subsequent pulse compression brings the amplified signal's frequencies back together. The pump supplies the added energy, rather than a stored population inversion.^[ref]

A. Dubietis, G. Jonušauskas and A. Piskarskas reported chirped and stretched pulse parametric amplification in beta barium borate in 1992. Their paper establishes a concrete early implementation of the parametric route; its published title identifies the nonlinear crystal.^[ref]

For pump, signal and idler angular frequencies, the conversion obeysEfficient amplification also requires phase matching, so contributions generated along the crystal reinforce. For collinear propagation the corresponding wave-vector relation is $k_p=k_s+k_i$; a noncollinear geometry requires the vector relation. Crystal dispersion and the chosen geometry determine which signal bandwidth is efficiently amplified.

An original energy-accounting example uses a 515-nanometre pump and an 800-nanometre signal. Their vacuum wavelengths giveSuppose 40 percent of a 2-millijoule pump is converted into signal–idler pairs, with negligible seed energy and other losses in this chosen model. Each conversion divides photon energy according to frequency, so the added signal energy isThe idler receives the remaining 0.285 millijoule. Assigning all depleted pump energy to the signal would violate this accounting.

If a compressor transmits 80 percent of that signal energy into a 50-femtosecond Gaussian pulse, the surviving energy is 0.412 millijoule. The chirped pulse amplification energy budget givesThese are stipulated teaching values, not a reconstruction of the 1992 experiment. Stretching lowers peak power during amplification; it does not remove the need to manage pump intensity or optical damage.

A useful apparatus check varies pump–signal delay while recording signal and idler spectra and energies. This tests temporal overlap and the frequency-conversion relation. Measuring the compressed pulse separately checks whether amplified bandwidth and residual spectral phase support the claimed pulse duration. Gain alone does not demonstrate successful compression, and an energy measurement alone does not establish peak power.

An optical parametric oscillator places an optical parametric amplifier inside an optical resonator that feeds back at least one generated wave. Pump energy is converted into signal and idler waves satisfying the frequency relation of the underlying three-wave mixing process.

Oscillation begins when round-trip gain overcomes loss at an allowed resonator mode. Below that threshold, weak generated emission can still occur, but sustained oscillation is absent. Tuning the phase matching and resonator together gives a tunable source at wavelengths difficult to obtain directly from a particular laser. Pump power alone does not select the output: cavity resonances, conversion gain and optical losses also matter.

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]

In quantum dense coding, a shared Bell pair lets Alice send two classical bits by transmitting one qubit. Use photon polarization as the computational basis, $|0⟩=|H⟩$, $|1⟩=|V⟩$, and define the Bell statesAlice and Bob initially share $|{\varphi }^{+}{⟩}_{AB}$. Alice encodes a two-bit message $(m,n)$ by applying the Pauli operator $X^n{Z}^m$ to her qubit:The last minus sign is a global phase. Alice transmits her photon to Bob, who now holds both qubits and uses a Bell-basis measurement to identify the message. The four Bell states are orthogonal, so an ideal Bell state analyzer decodes them without error. The resource cost is one previously shared Bell pair and one transmitted qubit per two classical bits; the shared quantum entanglement is consumed. Before transmission Bob's reduced density matrix is $I/2$ for every message, consistent with the no-communication theorem.

The BBO crystal supplies the polarization-entangled photon pair. An ultraviolet laser pumps this nonlinear optical crystal; spontaneous parametric down-conversion produces signal and idler photons with ${\omega }_p={\omega }_s+{\omega }_i$ and phase matching ${\mathbf{k}}_p={\mathbf{k}}_s+{\mathbf{k}}_i$. In a noncollinear type-II phase matching arrangement, collect the intersections of the two emission cones. The alternatives $|H{⟩}_A|V{⟩}_B$ and $|V{⟩}_A|H{⟩}_B$ must be indistinguishable in arrival time, spectrum and spatial mode. Compensation of birefringence and adjustment of the relative quantum phase then give $(|HV⟩+|VH⟩)/\sqrt{2}$; a local Pauli X gate converts it to $|{\varphi }^{+}⟩$. Half-wave plates and other waveplates implement Alice's Pauli gates. The BBO is the pair source; the message is encoded subsequently by the local optical elements.^[ref]

A basic passive linear optics Bell state analyzer with no extra resources identifies only two of the four polarization Bell states unambiguously. Thus the ideal two-bit capacity presupposes complete Bell-basis measurement; that basic optical implementation alone does not attain it deterministically.

The beta barium borate crystal supplies spontaneous parametric down-conversion: a weak ultraviolet pump can produce a pair of lower-frequency photons, with their frequencies and wavevectors selected by conservation of energy and phase matching. A suitably aligned and compensated source makes the two polarization alternatives coherent, giving the stated Bell state. The crystal alone does not guarantee that every collected pair has that state.

The two polarizing beam splitters route horizontal polarization and vertical polarization to different ports. Take $H$ from arm $a$ to the trigger $T$ and $V$ to arm $c$. The ordinary beam splitter in arm $b$ is a lossless, polarization-independent 50:50 splitter, with the unused input in the vacuum state. It splits probability amplitudes, rather than measuring which output a photon takes. The half-wave plate changes the relative phase of its principal polarizations by $\pi$; orient it so that a $V$ photon in $c$ becomes $(H+V)/\sqrt{2}$ in $c^{\prime }$, up to a common phase. Finally, photodetectors absorb arriving photons and supply electrical clicks. They need not resolve photon number for the ideal four-photon, fourfold-coincidence calculation: a click at every detector already requires one photon at each.

Choose the relative wave phases so that the final polarizing beam splitter sends $H_{c^{\prime }}$ to $H_2$, $V_{c^{\prime }}$ to $V_1$, $H_d$ to $H_1$ and $V_d$ to $V_2$. Such phase conventions must be used consistently: an uncompensated relative phase would give a phase-shifted GHZ state.

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.

Type-I phase matching requires $n_{e,2\omega }(\theta )=n_{o,\omega }$, because both pump photons are ordinary and the harmonic is extraordinary. Apply the direction-dependent index equation to the harmonic, not to the pump. Solving givesFor the ruby-laser data this is $0.626210$, whereas the shorter-wavelength data give $0.958873$. ThereforeThe second calculation uses the supplied $257\mathrm{nm}$ indices as rounded harmonic data: the exact half-wavelength of $515\mathrm{nm}$ is $257.5\mathrm{nm}$. The index dispersion between those two ultraviolet values was not supplied.

For an extraordinary wave, its Poynting vector makes an angle ${\theta }_S$ with the optic axis satisfying $\tan {\theta }_S=(n_o^2/n_e^2)\tan \theta$. Thus the spatial walk-off angle relative to the wave vector isBoth angles refer to the harmonic. The ordinary pump does not have this angular walk-off. For beam radius $w_b$, the approximate overlap length is $w_b/\tan \rho$, so the shorter-wavelength geometry has about $2.19$ times the walk-off-limited interaction length.^[ref]

To compare coupling, take propagation azimuth $\phi$ and ordinary unit polarization $(-\sin \phi ,\cos \phi ,0)$. Its quadratic source is $P_z=-{ϵ}_0{d}_{36}{E}_0^2\sin 2\phi$. Projection onto an extraordinary displacement direction with $z$ component proportional to $\sin \theta$ gives $|d_{\mathrm{eff}}|\propto |d_{36}\sin \theta \sin 2\phi |$, up to the small anisotropic normalization correction. The optimal azimuth is $45^{\circ }$. At equal beam area, interaction length, input power and approximately equal nonlinear coefficients, the shorter pump has both stronger angular coupling and the favorable frequency-squared factor:Here $\eta =P_{2\omega }/P_{\omega }$ in the small-conversion, equal-area model, with harmonic index phase-matched to the pump. Its smaller spatial walk-off further favors the $515\mathrm{nm}$ pump in a walk-off-limited geometry. Equal input power alone does not determine an absolute efficiency ranking: ultraviolet absorption, wavelength-dependent $d_{36}$, beam size, crystal length and coatings are unspecified. For example, a sufficiently strongly absorbing $257.5\mathrm{nm}$ output can reverse the ideal ranking. The stated preference is conditional on comparable transparent crystals and comparable focusing, not a prediction independent of those parameters.

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.