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Each mutually incoherent patch of an extended star produces a displaced double-slit interference pattern. The detector adds their light intensities. For two apertures separated by a projected interferometric baseline $D$, a source direction offset by a small angle ${\vartheta }_x$ changes their phase difference by $2\pi D{\vartheta }_x/\lambda$. The normalized sum over the source's angular brightness distribution is its first-order optical coherence, whose magnitude is the interferometric visibility for equally bright channels.

For a uniform circular stellar disk of full angular diameter $\alpha$, the normalized angular Fourier transform gives the uniform-disk stellar visibilityHere $J_1$ is the order-one Bessel function of the first kind. This follows by integrating $e^{i(2\pi D/\lambda ){\vartheta }_x}$ over the disk and dividing by its area. In particular, its power series gives $2J_1(x)/x=1-x^2/8+O(x^4)$.

For an ordinary slit separation $d=1\mathrm{mm}$, using the paper's $\alpha =22.6\times 10^{-8}\mathrm{rad}$ and observing wavelength $\lambda =570\mathrm{nm}$,Even $d=10\mathrm{mm}$ gives a visibility loss of only $1.94\times 10^{-5}$. Thus millimetre separation leaves the interference fringes almost indistinguishable from those of a point source; it gives very little sensitivity to the diameter.

The movable mirrors in the Michelson stellar interferometer collect light at a large adjustable separation $D$ and redirect it into closely spaced output beams. They increase the sampling interferometric baseline independently of the fixed output slit spacing $d$: $D$ sets sensitivity to source structure, whereas the output optics set the screen's interference fringe spacing. The first Bessel function zero is $x_1=3.83170597$, henceThe interference fringes disappear at this baseline even though each beam separately remains bright.

The value $3.08\mathrm{m}$ assumes a uniform disk. The original question supplies a diameter without a surface brightness law; real limb darkening changes the precise visibility zero. A one-dimensional uniformly bright strip of the same full angular width instead has a sinc function visibility and first zero $D=\lambda /\alpha =2.522\mathrm{m}$. That is a different source model, not the circular-disk result. The robust conclusion without a surface brightness model is that a baseline of a few metres, rather than millimetres, is required.

In SI units, the Maxwell equations in vacuum with prescribed charge density $\rho$ and current density $\mathbf{J}$ areWhen $\rho =0$ and $\mathbf{J}=0$, take the curl of the third equation and substitute the fourth. Using the curl of the curl identity and $\nabla \cdot \mathbf{E}=0$ gives the electromagnetic wave equation:For example, $\mathbf{E}=E_0\hat{\mathbf{x}}\cos (kz-\omega t)$, $\mathbf{B}=(E_0/c)\hat{\mathbf{y}}\cos (kz-\omega t)$ is a plane electromagnetic wave provided $\omega =ck$. The electric field, magnetic field and propagation direction are mutually perpendicular.

The Maxwell equations are linear, so the superposition principle applies to fields. However, the Poynting vector is quadratic:One may add the average wave intensities only when the cross terms vanish under the actual detector average. For co-propagating plane electromagnetic waves of parallel linear polarization, averaging over optical cycles but potentially resolving the difference frequency givesIf the detector integrates for $T$, the last term acquires $\mathrm{sinc}[({\omega }_1-{\omega }_2)T/2]$, with sinc function convention $\mathrm{sinc}x=\sin x/x$, evaluated at the midpoint phase. Thus $T\gg 1/|f_1-f_2|$ suppresses beats, and the infinite-time average adds the wave intensities exactly for distinct fixed frequencies. Random mutually uncorrelated wave phases also remove the cross terms in an ensemble average; orthogonal linear polarizations remove them for co-propagating waves in polarization-insensitive detection. Merely having different frequencies does not remove a resolved beat signal: two nearly equal frequencies with fixed relative wave phase are a concrete counterexample. For different propagation directions, the relevant addition is of average Poynting vectors, or their components through the chosen detector surface.

Take coordinate $y$ across the slit and observation angle $\theta$ from its normal vector. Under Fraunhofer diffraction, the relative optical path difference from $y$ is $y\sin \theta$. Uniform coherent illumination makes the complex amplitude proportional to the aperture integral:The light intensity is proportional to $|\mathcal{E}{|}^2$, soThe central value is the continuous limit of the sinc function. The first minima satisfy $\sin \theta =\pm \lambda /w$, if $\lambda \le w$. Infinite slit height removes spreading in the other transverse direction; this expression describes the slit-width direction.

Strictly, distance $D\gg w$ alone does not ensure Fraunhofer diffraction: one also needs $w^2/(\lambda D)\ll 1$, so that quadratic aperture-dependent wave phase is negligible, or observation in a suitable focal plane. For example, $w=1\mathrm{mm}$, $\lambda =500\mathrm{nm}$ and $D=1\mathrm{m}$ have $D/w=1000$ but $w^2/(\lambda D)=2$; the Fresnel diffraction correction is then substantial. The displayed result uses the intended far-field regime of the question and the usual scalar aperture approximation.

For an experimental-noise interpretation of Nyquist's theorem, a resistor $R$ in thermal equilibrium at temperature $T$ has one-sided open-circuit voltage-noise spectral densityin the classical range $hf\ll k_{B}T$, over an approximately rectangular bandwidth $\Delta f$. Across a matched load, half the open-circuit voltage appears, giving available noise power $k_{B}T\Delta f$. This is thermal agitation, present without a mean current, and is distinct from shot noise. A real readout weights the spectrum with its transfer function: $⟨v_{\mathrm{out}}^2⟩={\int }_0^{\infty }{S}_V(f)|H(f){|}^{2}df$. Cooling, reduced bandwidth and suitable impedance/noise matching reduce measured noise; the resistor's noise does not simply disappear in an ideal amplifier model.^[ref]

The bare name also has a sampling interpretation. The Nyquist–Shannon sampling theorem permits exact reconstruction of a bandlimited signal from sufficiently fast uniform samples. The practical condition is $f_s>2f_{\max }$, with an anti-aliasing filter before sampling; otherwise distinct continuous frequencies produce identical samples. Ideal reconstruction uses shifted sinc functions. Critical-rate equality needs an appropriate signal class and can fail for unrestricted edge-frequency sinusoids with unknown phase. Sampling fast enough prevents aliasing, but does not remove thermal noise. This distinction covers the source's unspecified use of “Nyquist's theorem”.^[ref]

Use the unitary position-to-momentum Fourier transform $\varphi (p)=(2\pi \hslash {)}^{-1/2}\int \psi (x)e^{-ipx/\hslash }dx$. Integration giveswhere $\mathrm{sinc}u=\sin u/u$ with its continuous value one at zero. The amplitude is real and even, with central maximum $\sqrt{a/(2\pi \hslash )}$, zeros at $p=2n\pi \hslash /a$ for nonzero integer $n$, and alternating-sign sidelobes decaying as $1/|p|$.

The probability density is $|\varphi {|}^2$, not the signed Sinc function amplitude. Its integral is one by the unitarity of the Fourier transform.

In this digital-sampling context, Nyquist's theorem means the Nyquist–Shannon sampling theorem. A continuous signal with no Fourier content outside $|f|\le B$ can be reconstructed from uniformly spaced samples taken sufficiently fast. The familiar practical condition isHere $f_N$ is the Nyquist frequency, while $2B$ is the Nyquist rate required by that signal bandwidth. A finite-energy bandlimited formulation allows equality; a rule intended to include arbitrary pure sinusoids at the boundary must be strict, because $\sin (\pi f_{s}t)$ vanishes at every sampling time $t=n/f_s$.

The mechanism is visible in the Fourier transform. Sampling by a time comb creates shifted copies of the original spectrum, separated by $f_s$:If those copies do not overlap, an ideal reconstruction filter selects one copy and recovers the original waveform. Equivalently, ideal interpolation uses the normalized sinc function:Each interpolation function is one at its own sample and zero at all other sample times. Exact reconstruction presumes the bandwidth condition, an ideal sampling clock and the full sample sequence.

If the spectrum extends beyond half the sampling rate, different continuous frequencies become indistinguishable: this is aliasing. A sinusoid at frequency $f$ is indistinguishable on the sampling grid from frequencies differing by integer multiples of $f_s$, with the usual sign reflection for real signals. For example, $10\mathrm{kHz}$ sampled at $8\mathrm{kS}{\mathrm{s}}^{-1}$ aliases to $2\mathrm{kHz}$; a later digital filter cannot tell which signal generated those samples.

Use an analogue anti-aliasing filter before the analogue-to-digital converter and choose a sampling rate leaving room for its finite transition band. Faster sampling can relax that analogue filtering problem, but simply increasing the number of samples in a fixed-duration record does not improve the independent Fourier-bin spacing beyond approximately $1/T$. A long recording is needed for low-frequency resolution, while a small sampling interval is needed for high-frequency coverage. Finite records also introduce windowing and spectral leakage.

Digitization adds amplitude quantization as well as time sampling. An ideal $N$-bit converter spanning voltage range $V_{\mathrm{span}}$ has step $\Delta =V_{\mathrm{span}}/2^N$. When quantization error may be modeled as independent and uniform across one step, its rms size is $\Delta /\sqrt{12}$; small coherent signals need not obey that white-noise approximation. Sampling rate controls temporal ambiguity, recording duration controls frequency resolution, and converter resolution controls amplitude discretization.

A Fourier series separates a periodic physical signal into harmonics. For example, a periodic voltage applied to a linear circuit can be treated one frequency at a time; the harmonic responses then superpose to give the output waveform.

Write $w=\Delta T$ for the pulse width and $\delta =w/T$ for its duty cycle, with $0<w<T$. The waveform is a train of positive rectangular pulses centred on $t=jT$, each with area one. It is even about $t=0$, so every sine coefficient vanishes. Cosine orthogonality givesand, for $n\ge 1$,This is the Fourier series of a periodic rectangular pulse:The actual constant term in the series is $a_0/2=1/T$, not $a_0$. As usual for a jump discontinuity, the series converges to the average of the one-sided values at each pulse edge, namely $1/(2w)$, irrespective of the value assigned at that isolated endpoint. Elsewhere it converges to the rectangular waveform.

Counting the constant as the first term, the first three terms areso their sum isIf “first three” instead means the first three nonconstant harmonics, add $a_3\cos (6\pi t/T)$; the original plot shows that alternative partial sum as well. The three-term approximation rounds the narrow peaks and ripples between them; many harmonics are needed to represent a sharp pulse.

The original sketches use the illustrative duty cycle $\delta =0.15$ and cover $-T\le t\le T$. The coefficient panel also displays the first spectral zero.

For $n\delta \ll 1$, expand the sinc function:Thus the low-index coefficients, including $a_0$, have nearly equal values, although their constant-series contribution still has the factor $1/2$. The first sinc zero occurs at $n\delta =1$, or $n=T/w$. If this is not an integer, the closest integer coefficient is near zero rather than exactly zero: writing $n\delta =1+\epsilon$ gives $a_n\simeq -(2/T)\epsilon$. This pulse-width scale of Fourier coefficients means that a narrower pulse requires significant harmonics extending to proportionally larger indices, the reciprocal time-width/frequency-width relation.

At first order in time-dependent perturbation theory set $c_1\simeq 1$ and use the parity result $V_{22}=0$. Decomposing the cosine into its two exponentials givesBoth exponentials contribute to the exact first-order result:Each apparent zero denominator has a continuous limit. In particular the near-resonant fraction tends to $it$ as $\omega \to {\omega }_0$, while the counter-rotating fraction remains bounded on the optical timescale. One may equivalently write the expression asThis form makes $c_2(0)=0$ explicit and avoids cancellation at resonance. At extremely short time both sinc functions tend to one, so $c_2\simeq -i\Omega t$, agreeing with the initial derivative of the original cosine-driven equation. No rotating-wave approximation has yet been made. The separate assumption of weak excitation remains necessary: the first-order expression must not be extended until it predicts probabilities comparable to or larger than one.

There are three independent nonfactorized momentum integrals, namely a tadpole integral, the three-line sunset diagram integral, and the five-line banana integral. The six displayed Feynman diagrams involve six complete integral expressions:The distinction between independent building blocks and their products removes an ambiguity in counting “different momentum integrals”. For the strict divergent-wave-function renormalization interpretation of ii, the graphs in c reduce to $T,T^2,B_5$, comprising three full integral expressions and two independent nonfactorized building blocks.

Make a Wick rotation and use $m>0$ to separate ultraviolet divergences from infrared divergences. Write $D_E(k)=1/(k^2+m^2)$, and let $T,B_r(P)$ be the Euclidean versions of the integrals above, with their harmless overall factors of $i$ removed. A spherical ultraviolet cutoff givesThus the one-vertex quartic graph is linearly divergent, and the one-vertex sextic graph is quadratically divergent as $T_{\Lambda }^2$.

For the other two building blocks, the spatial Fourier transform is especially transparent. The three-dimensional Euclidean scalar propagator isThe convolution theorem converts the $r$ parallel lines into $G(x{)}^r$. Angular integration gives, with a short-distance cutoff $a={\Lambda }^{-1}$,This is a different convenient cutoff regularization for the connected block; power-divergent coefficients depend on the cutoff convention, whereas the conclusions about powers and the presence of a kinetic logarithm do not. The sinc function is understood to be $1$ at $P=0$.

For $r=3$, the integrand starts as ${\rho }^{-1}+O(1)$, soIts divergent part is independent of external momentum. In particular,The finite difference and its finite derivative show why the pure three-line sunset diagram does not need a divergent wave-function renormalization in three dimensions.

For $r=5$, the small-distance Taylor expansion isConsequentlyIn particular, the nonzero $P^2\log \Lambda$ term requires a genuine divergent wave-function renormalization, in addition to the mass counterterm. The coefficients here refer to the displayed position-space cutoff and have not included Feynman-diagram symmetry factors or bare couplings.

The comparison with the superficial degree of divergence isThus the actual leading power agrees with $\omega$ in every case. The additional logarithms in products come from ultraviolet subdivergences, which uniform power counting alone does not resolve.

For $T{B}_3$ and $T^2{B}_3$, differentiation with respect to $P^2$ removes the logarithmic divergence of $B_3$, but leaves divergent factors $T$ or $T^2$ multiplying a finite, nonpolynomial function of $P^2$. These are subdivergent contributions, not new nonlocal counterterms: subtracting each tadpole diagram subdivergence and the momentum-independent $B_3$ subdivergence removes them. For example, in a factorized subtraction scheme $T{B}_3\mapsto (T-T_{\mathrm{div}})(B_3-B_{3,\mathrm{div}})$, and likewise for $T^2{B}_3$. The remaining $B_5$ kinetic logarithm explains the strict interpretation in ii. A dimensional regularization scheme can suppress power divergences; it does not invalidate this cutoff-based comparison with $\omega$.

For $0<k^2<4/{\gamma }^2$, defineThe simple poles are ${\omega }_{\pm }=\pm {\Omega }_k-i{a}_k$. In the inverse transform $\tilde{G}(k,t)=\int e^{-i\omega t}\hat{G}d\omega /(2\pi )$, close the contour below for $t>0$. Its clockwise orientation gives $-i$ times the sum of residues. That sum isFor $t<0$, close above; there are no poles. The residue theorem therefore givesHere $\Theta$ is the Heaviside step function. At $k=0$ the continuous retarded limit is $-t\Theta (t)$. At critical damping, ${\Omega }_k\to 0$ likewise replaces $\sin ({\Omega }_{k}t)/{\Omega }_k$ by $t$; beyond the stated oscillatory range an imaginary ${\Omega }_k$ converts it to a hyperbolic sine. These continuations retain decaying temporal exponents.

For each fixed $k$, taking $\gamma \to 0^{+}$ givesFor $t>0$, invert in space and set $s=kt$:The correctly normalized sinc function transform isFor instance, $\sin s/s=\frac{1}{2}{\int }_{-1}^1{e}^{isy}dy$, and Fourier inversion immediately gives the factor $\pi$. The PDF's prefactor $\sqrt{2/\pi }$ is inconsistent with its top hat of unit height; it would give height $\sqrt{2\pi }$. Using the normalized identity yieldsas a distribution. The values assigned precisely at a wave front do not change that distribution. This is the negative of the standard one-dimensional retarded wave Green function, because the PDF's differential operator is $-{\partial }_t^2+{\partial }_x^2$.

The sign and initial data can be checked without any contour calculation. Integrating the spatial-Fourier equation through the impulse givesIn real space these are $G_R(x,0^{+})=0$ and ${\partial }_t{G}_R(x,0^{+})=-\delta (x)$. In particular,for a smooth test function $f$, confirming the negative delta-function initial velocity. Before the impulse the displacement and velocity are zero. The disturbance fills the forward causal cone for the constant-speed wave equation, whose fronts move at unit speed, consistent with $\rho =\kappa =1$. The final question's printed $\rho =k$ is interpreted as $\rho =\kappa$, referring to the elastic constant, not to the Fourier wavevector. No initial conditions were otherwise specified, so the causal choice has been made explicitly.

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.

For the Fourier series of a periodic rectangular pulse, the sinc function is nearly one when $nw/T\ll 1$ and has its first zero at $nw/T=1$. Thus narrowing a pulse increases the range of significant Fourier coefficients. Only integer harmonic indices occur, so an envelope zero need not be an exactly vanishing coefficient.

For an ensemble with static detunings $\delta +\eta$, ideal same-phase quarter-turn pulses give ${\overline{p}}_2(T)=[1+\mathrm{Re}(e^{i\delta T}\chi (T))]/2$. Here $\chi$ is the characteristic function of the offset distribution. A mean-zero Gaussian of RMS width $\sigma$ gives $\chi =e^{-{\sigma }^2{T}^2/2}$; a symmetric uniform interval gives a sinc function. A width alone does not uniquely specify the envelope.

When the continuous Fourier transform $F$ of an integrable function vanishes outside $[-\pi ,\pi ]$, its periodic Fourier coefficients are the integer samples of the continuous representative of $f$, with the sign of the index reversed. Projecting $F$ in $L^2[-\pi ,\pi ]$ and then applying the Fourier inversion theorem gives the displayed reconstruction using the normalized sinc function, with each quotient defined as one when $t=n$. The Cauchy-Schwarz inequality controls the reconstruction error uniformly in $t$. Bessel's inequality bounds the vector of shifted sinc functions, giving absolute uniform tails when the sample sequence belongs to ${\ell }^2$.

A single transmitting aperture whose finite width produces diffraction. For a uniformly illuminated rectangular opening, its far-field amplitude is the Fourier transform of a constant over that opening, giving the sinc function pattern.

Single-slit diffraction is the spreading of light transmitted by one slit. Consider an ideal rectangular slit of width $a$, illuminated normally by a uniform plane wave with a single wavelength $\lambda$ in the surrounding medium. This monochromatic wave has uniform wave amplitude and equal incident wave phases across the slit, providing the required optical coherence. In the scalar Fraunhofer diffraction model, the normalized angular wave intensity isHere $\theta$ is measured from the forward normal in the plane across the slit width. The sinc function uses the unnormalized convention $\mathrm{sinc}u=\sin u/u$. The limit makes the central value one. This ideal envelope neglects additional direction-dependent propagation and detector factors; near the forward direction it describes the normalized screen irradiance. ^[ref].

The noncentral zeros satisfyThe restriction follows from the range of the sine of a real angle. The first minima have $\sin \theta =\pm \lambda /a$ when $\lambda /a\le 1$. If $a<\lambda$, this model has no noncentral zero in the forward hemisphere; equality puts the first zeros at grazing directions, outside the small-angle approximation and off any finite perpendicular screen. ^[ref].

For $0<\lambda /a<1$, define the central width as the separation between the two first minima. Its angular width and its width on a perpendicular screen areThe angular approximation uses radians and requires $\lambda /a\ll 1$; the screen approximation uses $y\approx L\theta$. The tangent relation is exact straight-line geometry, even though the underlying Fraunhofer diffraction field is approximate. This width is distinct from the full width at half maximum.

For $\lambda =500\mathrm{nm}$, $a=0.1\mathrm{mm}$ and $L=1\mathrm{m}$, the positive first minimum isThe dimensionless far-field parameter is $a^2/(\lambda L)=0.02$. Doubling the slit width to $a=0.2\mathrm{mm}$ gives $W\approx 5.00002\mathrm{mm}$ and parameter $0.08$; both curves below use the Fraunhofer diffraction approximation. Choosing $L=2\mathrm{m}$ halves those parameters and doubles both screen widths.

Each curve is normalized to its own $I_0$; the arrows measure between first minima. The calculation uses $\theta =\arctan (y/L)$ before evaluating the sinc function, rather than replacing $\sin \theta$ by $y/L$. The lower diagram shows the exact screen geometry with the vertical scale enlarged. Normalization removes the change in peak brightness when the slit width changes.

The wave interference explanation is especially simple: divide the slit into equal narrow strips and add their complex amplitudes, rather than their wave intensities. With wavenumber $k=2\pi /\lambda$, the continuous slit givesIntegrating the complex exponential gives the endpoint difference; its continuous value at zero is one. Taking the squared complex modulus gives the stated wave intensity. This is the Fourier transform of an interval indicator expressed in angular coordinates. Feynman's continuous-source argument obtains the same pattern by letting the number of sources tend to infinity. ^[ref].

At the positive first minimum, the wave phases span exactly $2\pi$ across the slit. Every strip in one half has a partner $a/2$ away, with equal wave amplitude and phase difference $\pi$, so the pairs cancel. At $\theta =0$, all the wave phases align instead. ^[ref].

The drawing uses $N=8$ equally weighted midpoint samples $x_j/a=-1/2+(j+1/2)/8$, with $j=0,\dots ,7$. At the first minimum their complex amplitudes are a common rotation of the eight roots of unity, so their head-to-tail polygon closes exactly. This finite sum illustrates cancellation; the continuous integral, rather than eight point sources, defines the slit pattern at general angles. The circle shows the continuous range of wave phases; coloured arrows show the eight samples.

A narrower aperture requires a larger angle to accumulate the cancelling phase difference, so narrowing the slit widens the beam. This is a useful limit on geometrical optics: confining light more tightly at an opening increases its subsequent angular spread. At fixed incident irradiance and slit height, reducing $a$ also reduces transmitted power, exposing a practical balance between diffraction-limited resolution and light throughput. In a diffraction grating, this single-slit diffraction envelope modulates the wave interference between different slits.

Spatial optical coherence compares field samples at different positions, usually across a transverse plane at equal time. Two small openings separated by $d$ sample this correlation: recombine their light with nearly matched travel times and measure interferometric visibility near the central interference fringes. A narrow opening in front of a broad-spectrum lamp can improve transverse spatial optical coherence while leaving its short temporal optical coherence. These are separate controls: narrow spectral selection by itself need not make an extended source spatially coherent. MIT's lecture notes illustrate the two-opening test and these contrasting source preparations. ^[ref].

To isolate source width, stipulate a distant, narrowband source with uniform angular brightness on $-\Theta /2\le \alpha \le \Theta /2$, and mutually uncorrelated emission from distinct source directions. Here $\Theta \ll 1$ is the full angular width in radians, $\lambda$ the common wavelength, and the two openings are separated by $d$ along the same transverse direction as the source extent. Neglect amplitude differences across the openings. Each source direction contributes a phase difference approximately $2\pi d\alpha /\lambda$. Averaging its complex cross term givesThe sinc function is extended continuously to one at $q=0$. The sign convention in the exponential depends on which opening is numbered first; the symmetric model gives the same real result either way. Equal detected single-opening light intensities make the magnitude equal to interferometric visibility. A displaced source center adds a deterministic wave phase, shifting the interference fringes without changing their contrast.

The strips show $I/(I_1+I_2)=1+g_{12}^{(1)}(0)\cos \varphi$ with equal single-opening light intensities, a common phase difference coordinate $\varphi$ and no single-slit diffraction envelope. At $q=1/2$, contrast is $2/\pi \approx 0.636620$; at $q=1$, it vanishes. At $q=3/2$, the correlation is $-2/(3\pi )\approx -0.212207$: weak fringes return with bright and dark positions exchanged. For $\lambda =500\mathrm{nm}$ and $d=0.5\mathrm{mm}$, the first zero occurs at $\Theta =1\mathrm{mrad}$. This is a calculated example, not an experimental measurement or a universal source-width law.

A proposed source-width control follows directly from this model: illuminate two fixed openings with a distant adjustable luminous slit of full width $w$ at distance $R$, so $\Theta \approx w/R$ under the small-angle approximation, and hold $d$ and the spectral selection fixed. Increasing $w$ overlays more mutually incoherent, displaced interference fringe patterns. Record profiles over several near-central fringes, measure each opening separately and correct intensity imbalance as in first-order optical coherence. Restrict the recorded region so the path difference stays small compared with the optical coherence length, and resolve the fringes with the detector. Real source nonuniformity, finite opening widths and spectral spread modify the stipulated sinc function law. This protocol is a model-based extension of the verified two-opening method, not a claim that the plotted curves were measured.

Thus an extended incoherent source can give locally excellent wave interference when the sampled separation is small, and almost none when it is larger. The useful scale $d\Theta \sim \lambda$ expresses how source extent limits the aperture over which coherent addition is available. The same angular averaging explains why narrowing a source opening can recover visible interference fringes, even without narrowing its optical spectrum.

The squared sinc function is nonnegative and integrable; it occurs as the Fourier transform of a triangular function.

It is the pointwise square of the sinc function, with its normalization inherited from that function.

A nonlinear optics process producing a field at the sum of two input angular frequencies. The driven electric polarization contains a term proportional to the product of the input complex amplitudes. With undepleted plane-wave inputs, integration along a crystal gives an output amplitude proportional to $L{e}^{i\Delta kL/2}\mathrm{sinc}(\Delta kL/2)$; the sinc function therefore determines the ideal length and phase mismatch dependence. A nonzero initial output field also interferes with this generated field.

A top-hat wavefunction is constant on an interval and zero elsewhere. It has a uniform position distribution and a sinc function momentum amplitude. Its discontinuous edges place it outside the finite-kinetic-energy domain of the free Hamiltonian. Smooth-edged localized packets can approximate its position profile while having different momentum tails.