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The Fourier expression is the scalar Fraunhofer diffraction approximation for coherent, effectively monochromatic illumination with a uniform incident plane wave over the aperture. At a distant screen the aperture-dependent quadratic propagation phase must be negligible, roughly $D^2/(\lambda z)\ll 1$ for transverse size $D$. Alternatively, a thin lens forms the angular Fourier transform in its back focal plane. That focal-plane construction is the appropriate one here; the aperture-to-lens distance need not itself be an enormous free-propagation distance. aperture features should be large compared with wavelength for this simple scalar transmission-mask model.

If complementary transmissions are $h$ and $1-h$, linearity gives $U_h+U_{1-h}=U_{\mathrm{free}}$. Away from the undiffracted forward beam, the uniform plane-wave contribution $U_{\mathrm{free}}$ vanishes in angular space, so $U_{1-h}=-U_h$ and their intensities agree. This is Babinet's principle. It equates the off-forward diffracted intensities, not the transmitted power or the on-axis intensity including the direct beam.

Let $X,Y$ be screen coordinates conjugate to aperture coordinates $x,y$, so $p\simeq kX/f$, $q\simeq kY/f$. With width $a=0.2\mathrm{mm}$ along $x$ and height $b=0.5\mathrm{mm}$ along $y$, define $\mathrm{sinc}u=\sin u/u$. The complementary rectangle has transformThus the obstruction's off-forward scattered component hasIts dark lines occur atThe central-lobe zero-to-zero size is $1.00\mathrm{mm}$ by $0.400\mathrm{mm}$; it is wider in the direction conjugate to the narrower obstruction. The central forward image of the illuminating beam must be distinguished from this scattered envelope in a real screen observation.

The source figure gives centre-to-centre separation $D=1.0\mathrm{mm}$. Choose the three rectangle centres at $x=-D,0,D$; an overall translation changes only the transform phase. Their scattered amplitude is proportional toThereforeAlong the screen's $x$ direction, the three-element interference factor has principal maxima separated by $\lambda f/D=0.100\mathrm{mm}$, under the single-rectangle envelope. Its zeros lie at $X=(m+1/3)(0.100\mathrm{mm})$ and $(m+2/3)(0.100\mathrm{mm})$. It also has weaker secondary maxima between principal orders. Every fifth principal order coincides with a single-rectangle zero and is missing; multiplication by the envelope slightly shifts maxima away from the nominal interference-factor positions. The figure plots the calculated scattered component and marks the separate direct-beam limitation.

Fresnel diffraction is interference during propagation when curvature of the optical path across an aperture matters. It provides a continuous account of fringes near geometric edges and their evolution into the far-field pattern. A screen that is many aperture widths away is not automatically in the Fraunhofer diffraction limit; wavelength also enters the comparison.

For plane-wave illumination of a transmission $h(\xi ,\eta )$ at $z=0$, the Huygens-Fresnel principle in the scalar paraxial approximation givesIt follows by expanding the propagation distance as $z+[(X-\xi {)}^2+(Y-\eta {)}^2]/(2z)$ while treating the amplitude prefactor as slowly varying. Small propagation angles justify dropping higher path terms. Finite coherent illumination, vector edge fields and very small apertures can require a more complete theory. The Fresnel number $N_F=a^2/(\lambda z)$, with a clearly declared aperture scale $a$, measures whether the aperture's quadratic phase is significant. When it is negligible, the remaining linear phase produces the aperture Fourier transform.^[ref]

Define the optical Fresnel integralsTheir limiting values are $C(\infty )=S(\infty )=1/2$, with odd continuation to negative arguments. The Cornu spiral plots $(C(u),S(u))$: the field contributed by a finite interval is proportional to the chord between its endpoints. Intensities require squared chord lengths; summing zone intensities would lose the cancellation mechanism.

For an opaque straight edge transmitting $\xi >0$ and all $\eta$, let $v=X\sqrt{2/(\lambda z)}$, so $X>0$ is the geometrically illuminated side. Integrating in the unrestricted transverse direction and using the half-line in the other giveshenceFar into the shadow, $I\to 0$; far into the illuminated region, $I\to I_{\mathrm{in}}$. At the geometric shadow boundary, $I(0)=I_{\mathrm{in}}/4$, not half the incident intensity: the field is halved there. On the illuminated side, alternating constructive and destructive contributions give bright and dark fringes whose contrast declines away from the edge. Their positions scale with $\sqrt{\lambda z}$ and are not equally spaced. A finite detector averages fine fringes and tends to recover the geometric-optics step. In a controlled experiment, translate a straight edge through a broad collimated beam, scan a small photodetector across the screen and normalize against an unobstructed reference; vary $z$ to test the square-root scaling. Source size, spectral bandwidth, edge roughness and detector aperture wash out contrast and must be distinguished from a failure of the propagation law.

For a rectangular aperture of full widths $a,b$, defineThe two transverse integrals separate:andA long slit leaves only one varying Fresnel factor. For a square centred on axis, $a=b$ and $w=(a/2)\sqrt{2/(\lambda z)}$ giveOpposite edge contributions overlap; changing aperture size or propagation distance changes their relative phase. Consequently a geometrically illuminated centre can brighten or darken, rather than remaining a fixed top-hat plateau. The original rectangular-aperture map shows fringes parallel to both edges and their crossing structure. Outside the geometric rectangle the wave also extends into the shadow.

At large Fresnel number an interior region approaches geometric illumination while edge fringes remain on a scale $\sqrt{\lambda z}$. At intermediate values the opposite-edge fringes interact across the whole aperture image. In the far field, the formula tends to the product of sinc-squared factors,The far-field lobe scale $\lambda z/a$ grows linearly with $z$, unlike the local near-edge scale. The transition is controlled by the neglected phase, not an abrupt physical boundary.

To test the main effect, record normalized profiles at several distances, calculate the Fresnel integral with the measured aperture dimensions and compare fringe positions before fitting a free intensity scale. Use the same transverse coordinate origin and check that the incident beam is sufficiently uniform. Babinet's principle relates complementary aperture and obstruction fields through $U_a+U_o=U_{\mathrm{free}}$; their near-field intensities are not generally equal, because the unobstructed field remains present and interferes with the diffracted field. This is why a far-field obstruction argument cannot simply be carried over as equality of Fresnel intensities.

Distinguishable noninteracting particles. Interpret $V$ as the same external one-particle potential for each particle; the total potential is $V({\mathbf{r}}_1)+V({\mathbf{r}}_2)$. The Time-dependent Schrödinger equation isLet $\hat{h}{\varphi }_a={ϵ}_a{\varphi }_a$ and $\hat{h}{\varphi }_b={ϵ}_b{\varphi }_b$, where $\hat{h}=-{\hslash }^2{\nabla }^2/(2m)+V$. Applying ${\hat{h}}_1+{\hat{h}}_2$ to the product multiplies it by ${ϵ}_a+{ϵ}_b$. The time derivative of $e^{-iEt/\hslash }$ multiplies it by $E$. Thus the proposed stationary product is a solution providedThe tensor product separates the two independent one-particle dynamics.

Identical spin-one-half particles. Identical fermions require antisymmetry under exchange of the complete position-and-spin labels. First take $a,b$ to be distinct orthonormal spatial orbitals, and defineThe symmetric spatial state must multiply the antisymmetric spin-one-half singlet state; the antisymmetric spatial state must multiply one of the three symmetric spin-one-half triplet states. The four normalized possibilities, each with the common time factor $e^{-iEt/\hslash }$, areThese are the required product states under fermionic exchange symmetry. If $a=b$, the antisymmetric spatial combination vanishes. Only the product ${\varphi }_a({\mathbf{r}}_1){\varphi }_a({\mathbf{r}}_2)$ times the singlet remains, as required by the Pauli exclusion principle. A nonorthogonal pair would require different normalization factors; the displayed $1/\sqrt{2}$ assumes an orthonormal stationary-state basis.

Weak repulsive Coulomb interaction. For fixed orbitals $a,b$, first-order perturbation theory gives spatial energy shifts $J\pm K$. Here $J$ is the direct interaction energy, while the exchange integral for repulsion strength $g>0$ isThe positivity follows from the positive Fourier transform of the Coulomb kernel. Consequently Coulomb exchange splitting of two orbitals givesAll three triplet spin components remain degenerate for a spin-independent Coulomb interaction. Their antisymmetric spatial factor vanishes at coincidence and reduces close encounters. The inequality is strict when $K>0$; for negligible orbital overlap, the splitting may vanish. This comparison fixes one particle in each of the specified orbitals and does not rank singlets with a different orbital occupation.

Spin-zero bosons. For one particle in each of the same distinct orthonormal orbitals, the spin factor is symmetric and bosonic exchange symmetry requires ${\Phi }_{+}$. At coincident positions,Thus the bosonic coincidence enhancement in two orthogonal orbitals isThese are joint probability densities, or small-detector-volume limits; an exactly specified point has zero probability in a continuous distribution. If both particles occupy the same orbital, the properly normalized product is already symmetric and the corresponding enhancement relative to distinguishable particles in that same orbital is one. The paper's usual factor-two answer presumes distinct orthogonal input orbitals.

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.

Superpose two nearby harmonic waves. The identityseparates a rapid carrier from a slow beat envelope. A constant point of the envelope moves at $\Delta \omega /\Delta k$, which tends toas the components approach one another. Equivalently, expand a narrow wave packet spectrum around $k_0$: $\omega (k)={\omega }_0+v_g(k-k_0)+\cdots$. Factoring the carrier out of its Fourier transform representation leaves an envelope depending on $x-v_{g}t$. This interpretation requires a narrow spectrum and little attenuation or distortion; higher derivatives of the dispersion relation can change the envelope shape. The carrier phase velocity is $v_p=\omega /k$ and need not equal $v_g$.

Use Fourier transform convention $\hat{f}(q)=\int f(x)e^{-iqx}dx$. The convolution theorem givesA finite array of very narrow slits at $x=jD$, $j=0,\dots ,N-1$, has transmission proportional to ${\sum }_j\delta (x-jD)$. It can be constructed by multiplying an infinite Dirac comb by a finite window. The infinite comb transforms to a reciprocal comb; the finite window convolves each reciprocal peak with a narrow sinc-shaped amplitude. Thus increasing the illuminated length narrows each Fraunhofer diffraction peak.

The exact finite sum makes the argument quantitative:Its normalized intensity is $[\sin (NqD/2)/(N\sin (qD/2)){]}^2$. Principal maxima occur at $qD=2\pi m$; their first zeros are displaced by $|\Delta (qD)|=2\pi /N$. Replacing the ideal narrow slit by a finite slit function convolves the array in real space, multiplying this field by the single-slit Fourier transform. Field amplitudes are transformed and combined before their squared modulus is taken.

Circular apertures and circular obstructions illustrate both the usefulness and the limits of geometric shadow arguments. In scalar diffraction theory, each point of a transmitted wavefront contributes a complex secondary field. Their coherent sum, described by the Huygens-Fresnel principle, determines the intensity. Circular symmetry simplifies the integral, but the answer depends on whether propagation is in the Fraunhofer diffraction or Fresnel diffraction regime.

For aperture radius $a$ and axial propagation distance $z$, the Fresnel number $N_F=a^2/(\lambda z)$ measures the importance of quadratic phase across the aperture. The far-field condition is $N_F\ll 1$; a lens can produce the same angular spectrum in its back focal plane. Fresnel propagation retains that quadratic phase. For an axial point source a distance $s$ before the object and an observer a distance $r$ after it, the phase uses the effective distance $z_{\mathrm{eff}}=sr/(s+r)$, because both incident and outgoing spherical paths contribute curvature. Scalar, paraxial propagation and an opening large compared with wavelength are assumed throughout.

A circular hole in the far field gives an Airy pattern. For uniform plane illumination the field is the two-dimensional Fourier transform of the aperture. In polar coordinates, let $Q=k\sin \theta$; the angular integral gives the Bessel function $J_0$:ThusThe limiting central value is one because $J_1(u)\simeq u/2$ near zero. A bright central Airy disk is surrounded by progressively weaker concentric rings. The first dark ring has $ka\sin \theta =3.8317$, or $\sin \theta \simeq 1.22\lambda /D$ with diameter $D=2a$. This scale gives the Rayleigh criterion for a circular optical instrument. Larger apertures narrow the angular pattern; a lens of focal length $f$ maps the first dark ring to radius $1.22\lambda f/D$ in its focal plane.

In the Fresnel regime, the axial intensity oscillates as aperture size or distance changes. Fresnel half-period zones have boundaries at path excesses $m\lambda /2$ and approximately opposite phases in successive zones. For a plane wave, their radii obey $r_m^2=m\lambda z$. A circular opening can therefore expose alternating constructive and cancelling zone contributions. More precisely, Fresnel diffraction by a circular aperture gives, normalized by the unobstructed field,An odd integer number of zones gives a bright axial point with ideal intensity $4I_0$; an even number gives an axial zero. Intensity can locally exceed the unobstructed value because interference redistributes energy; total transmitted power still obeys conservation of energy. The relative normalisation here differs from the Airy expression, which normalises each far-field pattern by its own central intensity.

Off axis, the circular Fresnel field isIts concentric structure evolves with $N_F$ and includes fringes near the geometrical edge. The axial zone argument does not determine the complete off-axis pattern.

Complementary holes and opaque disks are related at the field level. Babinet's principle states $U_a+U_d=U_0$ for complementary aperture and disk transmissions under the same illumination. In the Fraunhofer region away from the direct transmitted beam, the background field is absent and the diffracted fields are opposite in sign. Their scattered intensity profiles therefore have the same Airy angular dependence. Exactly in the forward direction the unobstructed field must be retained; subtracting intensities is incorrect. Finite incident beams also give a finite-width direct beam rather than an ideal angular delta function.

For Fresnel diffraction by an opaque disk, the axial complement is particularly striking:The ideal circular disk has an Arago spot at the centre of its geometric shadow. Circular edge contributions reach the axial point with the same phase. This does not mean the whole shadow is illuminated: off-axis cancellation produces the surrounding shadow and rings, and the axial spot becomes narrower for larger obstruction or shorter propagation distance. An aperture's axial intensity oscillates, while its complementary disk's axial intensity stays constant in this ideal paraxial model.

A practical observation uses a compact coherent source, an accurately circular hole or disk, and an aligned screen or camera. Varying the propagation distance or aperture radius reveals the transition between regimes. A point source, or spatial filtering, prevents different source positions from washing out the fine structure. Finite source size, incomplete coherence, edge roughness, misalignment, vector effects for wavelength-sized objects and lens aberrations reduce the ideal contrast. These experimental limitations explain why a geometrically unexpected axial bright point requires a suitable setup even though its scalar derivation is simple.

Let the aperture occupy the $z=0$ plane, with coordinates $(x,y)$ and opening $\Sigma$. Write ${\psi }_{\Sigma }(x,y)$ for the incident scalar complex amplitude and $h(x,y)$ for its dimensionless amplitude transmission, including any imposed phase. The observation point has transverse coordinates $(X,Y)$, corresponding to the PDF's $(x_0,y_0)$, and axial distance $R$. Also $k=2\pi /\lambda$ is the wavenumber and $i^2=-1$.

The Huygens-Fresnel principle expresses the propagated field, up to common prefactors and an approximately constant obliquity factor, asIn the paraxial approximation, expand the path asThe last term is negligible in the Fraunhofer diffraction limit, while the first two terms and $1/r\simeq 1/R$ are common across the aperture. Factoring them out leavesIt is the two-dimensional Fourier transform of the transmitted aperture field, evaluated at transverse wavevector $(kX/R,kY/R)$. For appreciable observation angles the corresponding exact directional components are used rather than $X/R,Y/R$.

For mutually incoherent spectral components, the detected intensities add. If $S(\nu )$ describes the mean output spectral intensity, the interferogram isSubtract the constant background, scan the mirror with a calibrated displacement, and take the Fourier transform with respect to delay $\tau =x/c=2d/c$. This is Fourier transform spectroscopy. The transform recovers the spectrum subject to detector response, finite scan length and any window function. Sampling the displacement finely enough to resolve the shortest optical fringes avoids aliasing.

Using the usual reciprocal-maximum-delay resolution estimate, $\delta \nu \simeq c/x_{\max }$, and $|\delta \nu |\simeq c\delta \lambda /{\lambda }^2$, givesThe mirror displacement is measured from zero optical path difference. This delay-range resolution in Fourier transform spectroscopy is a conventional estimate: a specified window and resolution definition introduce a numerical factor. For example, a symmetric rectangular delay window has its first instrumental zero at $\delta \nu =1/(2{\tau }_{\max })$, which would give $6.25\mathrm{mm}$ under that particular criterion. The paper does not specify a line-profile criterion; the standard $12.5\mathrm{mm}$ estimate states its convention explicitly.

The convolution of two aperture functions in one dimension isWith Fourier transform convention $\hat{f}(q)=\int f(x)e^{-iqx}dx$, the convolution theorem statesThe first identity follows by substituting the convolution integral, putting $v=x-u$, and separating the resulting double integral into $\int f(u)e^{-iqu}du$ times $\int g(v)e^{-iqv}dv$. For an absolutely integrable pair the interchange is justified by absolute convergence; finite optical apertures satisfy suitable integrability assumptions.

In Fraunhofer diffraction, the observed field is the Fourier transform of the transmitted complex aperture field. For two identical slits, let $g(x)$ be one slit and use a pair of Dirac delta distributions to locate the centres:Consequently $\hat{t}(q)=2\hat{g}(q)\cos (qD/2)$. For a rectangular slit, $\hat{g}=a\mathrm{sinc}(qa/2)$, giving a single-slit diffraction envelope times the double-slit interference factor in intensity. A $\pi$ phase reversal replaces the plus between the deltas by a minus, giving $\sin (qD/2)$ instead, as in B7.

More generally, an $N$-slit array is a single-aperture function convolved with a finite comb. Its field contains the factorThis separates the narrow diffraction grating maxima from the individual-aperture envelope. The same reasoning holds in two transverse dimensions. Multiplying an existing aperture by an additional finite mask instead convolves their field transforms and can broaden features. These are operations on complex fields: intensity is the squared modulus of the final field and cannot generally be substituted into the field-level convolution theorem.

A localized quantum particle is described by a wave packet: a coherent quantum superposition of momentum components, rather than a single delocalized plane wave. The Born rule assigns the position probability density $|\psi (x,t){|}^2$. A useful normalized initial Gaussian wave packet isIts mean position and momentum are $x_0$ and $p_0$, and its standard deviations satisfy ${\sigma }_x(0)={\sigma }_0$, ${\sigma }_p=\hslash /(2{\sigma }_0)$. This unchirped initial state saturates the Heisenberg uncertainty relation.

For a free particle, the momentum-space Schrödinger equation multiplies each amplitude by $e^{-i{p}^{2}t/(2m\hslash )}$. The inverse Fourier transform is a Gaussian integral, giving the free Gaussian wave packet evolutionOne way to obtain this expression explicitly is to write the initial momentum amplitude as proportional to $\exp [-{\sigma }_0^2(p-p_0{)}^2/{\hslash }^2-ip{x}_0/\hslash ]$, multiply by the free evolution phase and complete the square in $p$. Taking the modulus squared removes the position-dependent phase and leavesThe expectation value follows uniform classical motion while the density spreads. A narrower initial packet needs a wider momentum distribution and therefore spreads more rapidly. At long times its width grows linearly with $|t|$; its variance grows quadratically, unlike the linear variance growth of ordinary diffusion. The probability continuity equation preserves total probability despite the decreasing peak height. The mean momentum and the mean energy $(p_0^2+{\sigma }_p^2)/(2m)$ remain constant, since the momentum populations do not change.

The same result has a wave interpretation. For a free nonrelativistic particle $\omega (k)=\hslash k^2/(2m)$, so the group velocity is $d\omega /dk=\hslash k/m$, equal to $p_0/m$ at the packet center. The curvature $d^2\omega /d{k}^2=\hslash /m$ makes different momentum components travel with different group velocities, producing dispersion. The phase velocity is $\omega /k=\hslash k/(2m)$ for this energy convention and does not describe the motion of the envelope. The evolving quadratic spatial phase is essential to the spreading; a density snapshot alone does not specify the full quantum state.

A freely spreading Gaussian wave packet remains a pure Gaussian state but develops position-momentum correlations. Its symmetrized covariance is $C_{xp}=⟨\{x-⟨x⟩,p-⟨p⟩\}⟩/2=t{\sigma }_p^2/m$. ConsequentlyEquality in the Robertson-Schrödinger uncertainty relation survives at every time, whereas the simpler product ${\sigma }_x{\sigma }_p$ equals $\hslash /2$ only initially. A chirped Gaussian wave packet can already have $C_{xp}(0)\ne 0$. More generally, $x(t)=x(0)+p(0)t/m$ gives ${\sigma }_x(t{)}^2={\sigma }_x(0{)}^2+2t{C}_{xp}(0)/m+t^2{\sigma }_p(0{)}^2/m^2$. A sufficiently negative initial covariance permits initial contraction before subsequent spreading.

A potential energy changes this behavior. A quadratic potential preserves the Gaussian form because the evolution of position and momentum is linear. For a quantum harmonic oscillator, the Gaussian covariance in a harmonic potential isA coherent state has ${\sigma }_x^2=\hslash /(2m\omega )$, ${\sigma }_p^2=m\hslash \omega /2$ and zero centered covariance. Its width is constant, and its center follows $x_c(t)=x_0\cos \omega t+p_0\sin \omega t/(m\omega )$. In oscillator notation its coherent state label rotates as $\alpha (t)=\alpha (0)e^{-i\omega t}$, apart from the common ground-energy phase of the state vector. A squeezed coherent state generally has a width that breathes at $2\omega$. Thus being Gaussian does not imply either free spreading or constant width; the Hamiltonian operator and initial covariance determine the result.

For a general nonquadratic potential, different parts of the packet experience different nonlinear forces, and an initially Gaussian profile need not remain Gaussian. Ehrenfest theorem gives the exact relations $d⟨x⟩/dt=⟨p⟩/m$ and $d⟨p⟩/dt=-⟨V^{\prime }(x)⟩$. Replacing the last mean by $-V^{\prime }(⟨x⟩)$ is exact for a quadratic potential and approximate for a packet narrow compared with the scale on which the force varies. This connects packet motion to the correspondence principle without treating a quantum state as a definite classical path. Scattering can split a packet into reflected and transmitted components, while confinement can produce oscillations and interference between discrete energy eigenstates. Their relative phases evolve even when their populations remain constant. Localization, spreading and coherent motion all follow from the same linear unitary time evolution.

In Fresnel diffraction, the propagation wave phase varies quadratically across the aperture and that curvature must be retained. For a transverse scale $w$ at propagation distance $L$, its size is of order $\pi w^2/(\lambda L)$. In Fraunhofer diffraction, that quadratic term is negligible, approximately $w^2/(\lambda L)\ll 1$, while the linear phase encoding the observation angle remains. A lens can also form the far-field angular pattern in its focal plane.

For a normally incident plane wave and aperture transmission $h(x,y)$, the far-field complex amplitude is its Fourier transform:The detected light intensity is proportional to $|\psi {|}^2$. For identical parallel slits, write the transmission as a sum of shifted copies of one slit. The Fourier shift theorem gives one single-slit transform multiplied by the sum of positional phase factors:The single-slit diffraction envelope sets the broad scale, while coherent interference between slit centres produces the finer array pattern. One sums complex amplitudes before squaring, rather than adding independent slit intensities.

A nondispersive wave has a linear dispersion relation $\omega =vk$ over the relevant range, so every Fourier transform component has the same phase velocity $v_p=\omega /k=v$ and group velocity $v_g=d\omega /dk=v$. A wave packet then translates without spreading in an ideal homogeneous lossless medium.

For wave dispersion, $v_p$ depends on frequency or wavenumber. Different components accumulate different phases, and in general the group velocity differs from the phase velocity; variation of group velocity across the spectrum produces spreading of a wave packet. The membrane material can be nondispersive while confinement in a membrane waveguide introduces wave dispersion along its length.

For the normally incident plane wave, the incident complex amplitude and phase are uniform across the aperture. Treat the field using scalar diffraction, with the aperture represented by its prescribed amplitude transmittance $h(x,y)$. Let $R$ be the distance from the aperture centre to the observation point and let ${\ell }_x,{\ell }_y$ be the transverse direction cosines of that point.

Across an aperture of characteristic size $a$, the Fraunhofer diffraction assumptions are that $a/R\ll 1$, so $1/r\simeq 1/R$ and the obliquity factor is nearly constant, and that the quadratic contribution to the propagation phase is negligible:The remaining path-length dependence is linear:A coherent monochromatic wave and an aperture large enough relative to wavelength for the scalar aperture model complete the usual physical assumptions. The complex amplitude becomeswhere $C$ includes the common propagation phase, distance factor and nearly constant obliquity factor. For the central part of a planar screen in the paraxial approximation, ${\ell }_x\simeq X/R$ and ${\ell }_y\simeq Y/R$. Small observation angles are useful for that screen-coordinate formula; a general far-field Fourier transform can instead use the exact direction cosines. A lens can also produce this pattern in its focal plane without requiring a physically very distant screen.

The far-field complex amplitude is the two-dimensional Fourier transform of the aperture amplitude transmittance, evaluated at transverse spatial frequencies $q_x=k{\ell }_x$, $q_y=k{\ell }_y$. The light intensity is proportional to the squared magnitude of this transform.

Use linearity to add the transforms of simpler aperture components, preserving their relative phases. A translated component acquires a complex exponential factor. Repeated identical openings can be expressed as the convolution of one opening with a sum of Dirac delta functions; the convolution theorem then factors the field into a single-opening transform and an array factor. Multiplying aperture functions instead convolves their transforms. These operations simplify complex apertures without repeatedly evaluating the full diffraction integral. It is the complex amplitudes, rather than their wave intensities, that are added for coherent illumination.

Expand the propagation distance at small angles:The first two terms supply an aperture-independent phase. If the aperture quadratic phase is small, $k(x^2+y^2)/(2L)\ll 1$, it too may be dropped from the integrand. The remaining Fraunhofer diffraction amplitude is the aperture's Fourier transform at transverse wavevector $(k{x}_0/L,k{y}_0/L)$, as required.

For slit width $w$ and center spacing $d$, adding the two shifted top-hat transforms gives the normalized intensityThe aperture outer half-width is $17.5\mu \mathrm{m}$; its maximum quadratic phase is only about $1.5\times 10^{-3}$ radians at the given wavelength and distance, so the Fraunhofer approximation is excellent. Fringe spacing is $\lambda L/d=25.28\mathrm{mm}$ and the first single-slit envelope zero is at $\lambda L/w=63.2\mathrm{mm}$.

The side maximum of a finite-width double-slit pattern shifts slightly inward from $25.28\mathrm{mm}$. Differentiating the full intensity gives, away from zeros,Its first side maximum is at $X\simeq 23.88\mathrm{mm}$ with $I/I(0)\simeq 0.592$. The expected rate is therefore $59.2{\mathrm{s}}^{-1}$. Poisson shot noise gives signal-to-noise ratio $N/\sqrt{N}=\sqrt{N}$, so one needs $N=100$ counts:Using the common fringe-center approximation instead gives relative intensity ${\mathrm{sinc}}^2(\pi w/d)\simeq 0.573$ and $t\simeq 1.75\mathrm{s}$. Both express the same order of exposure; the first uses the true maximum. Dark counts, background subtraction and uncertainty in the reference rate are excluded by the ideal stated noise model.

Fresnel diffraction retains quadratic aperture-dependent propagation phases and generally depends on both aperture and screen distance. Fraunhofer diffraction neglects those phases and gives the Fourier transform of the aperture in angle space, either sufficiently far away or in a suitable lens's focal plane.

The calculations below use the Fraunhofer, paraxial approximation. For an overall aperture scale $D\sim 1000\lambda$, free propagation requires $L\gg D^2/\lambda \sim 10^6\lambda$. The printed $L\gg 1000\lambda$ alone only makes the separation small relative to distance; it is not by itself the full far-field condition.

A Michelson interferometer uses a beam splitter to divide collimated input light between a fixed mirror and a moving mirror. Returning beams recombine at the beam splitter; one output reaches a photodetector. A compensator plate can equalize the glass traversed in the two arms. Moving one mirror by $x$ changes the optical path difference by $2x$.

The usual nominal Fourier transform spectroscopy specification is $\delta \nu \simeq 1/d_{\max }$, where $d_{\max }$ is the maximum optical path difference measured from equal arms. It givesThis is a nominal spectral resolution element convention.^[ref] A symmetric rectangular scan from $-d_{\max }$ to $d_{\max }$ instead has a sinc instrumental line shape with first-zero offset $1/(2d_{\max })$. Quoting that first-zero criterion would give $d_{\max }=1$ cm; interferogram apodization and other linewidth criteria change the numerical factor. An exact mathematical minimum requires such a criterion.

In the measurement chain, the Michelson interferometer is the spectral analyzer or modulator, between the collection optics and the photodetector. It encodes the incoming radiation spectrum into an interferogram; detector electronics digitize that signal and a Fourier transform reconstructs the radiation spectrum. It does not itself turn radiant power into an electrical signal.

A dominated differentiation under the integral sign suffices. Suppose $\varphi (\cdot ,t)$ is measurable, $\varphi (\cdot ,t_0)$ is integrable at one parameter value, and for almost every $x$ the map $t\mapsto \varphi (x,t)$ is continuously differentiable on an interval. If $|{\partial }_t\varphi (x,t)|\le H(x)$ throughout that interval for an integrable $H$, then the parameter integral is differentiable, andOne direct sufficient version assumes ${\partial }_t\varphi (x,t)$ is continuous in $t$ almost everywhere and has that common domination; dominated convergence theorem then proves the assertion.

Here $|f(x)e^{itx}|=|f(x)|$, so the complex-valued function $h$ is integrable for every real $t$. For a real increment $u\ne 0$, the elementary bound $|e^{iux}-1|\le |ux|$ givesThe right side is integrable by hypothesis and the integrand converges pointwise to $ixf(x)e^{itx}$. dominated convergence theorem, applied to real and imaginary parts, provesThis supplies the domination explicitly rather than merely differentiating a formal Fourier transform.

Noninteracting spectral line. Treat $\mathbf{k}$ as a wavevector, so physical crystal momentum is $\hslash \mathbf{k}$. Write ${\Omega }_{\mathbf{k}}={\epsilon }_{\mathbf{k}}/\hslash$; an energy zero at the chemical potential can be chosen throughout. The temporal factor of a noninteracting state is $e^{-i{\Omega }_{\mathbf{k}}t}$. With the Fourier transform convention $\tilde{\psi }(\omega )=\int dt{e}^{i\omega t}\psi (t)$,Thus its normalized frequency spectral function isOne should not square the delta distribution as if it were an ordinary finite-time transition amplitude. More precisely, the fermionic retarded Green function is $G_0^R(\mathbf{k},t)=-i\Theta (t)e^{-i{\Omega }_{\mathbf{k}}t}$. Its Fourier transform is $(\omega -{\Omega }_{\mathbf{k}}+i{0}^{+}{)}^{-1}$, whose negative imaginary part divided by $\pi$ is the same delta peak.

Interactions and lifetime. Interactions change the dispersion through the real part of the self-energy and permit decay into other excitations through its imaginary part. In the simplest quasiparticle-pole model, exponential decay of the retarded amplitude givesPerforming the causal integral,The corresponding Lorentzian quasiparticle spectral peak isIts total frequency integral is one. The exponential amplitude lifetime is $1/{\Gamma }_{\mathbf{k}}$; the probability lifetime is $1/(2{\Gamma }_{\mathbf{k}})$. Describing ${\Gamma }_{\mathbf{k}}$ as a scattering rate therefore requires the lifetime convention to be specified. The energy half-width is $\hslash {\Gamma }_{\mathbf{k}}$.