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In linear scalar diffraction theory, complementary aperture transmissions $t$ and $1-t$ give complex fields $U_t+U_{1-t}=U_0$, where $U_0$ is the unobstructed field under identical illumination. Away from the direct beam in Fraunhofer diffraction, their diffracted fields differ by a sign and have equal intensity. Intensities do not generally add: in particular, complementary on-axis Fresnel diffraction patterns need not have equal intensity. The ideal scalar statement does not incorporate vector boundary details at a real conducting edge.

An actual teaching setup is described in Laboratory Exercise No. 4: Diffraction, MIT 6.237, Fall 2024, for MIT's 6.237 Modern Optics Project Laboratory, Fall 2024. Preparation includes predicting the aperture-size dependence and the distinction between Fraunhofer diffraction and Fresnel diffraction. A helium–neon laser supplies $632.8\mathrm{nm}$ light; the far-field setup uses a planar incident wavefront, a diffracting object and a distant screen, with beam expansion when needed. Figure 1 also marks an optional CCD camera. Exercise 4.7 asks students to sketch the circular-aperture pattern, measure the Airy disk diameter between correctly identified boundaries, and infer the aperture size. No measured values are supplied. As a regime comparison, exercise 4.9 varies the circular opening's position near a microscope objective optical lens, observes near-field patterns and compares them with the far-field case. This tests whether a change in propagation geometry changes the pattern. The coherent illumination measures one aperture's response, rather than the discrimination of two mutually incoherent optical point sources.^[ref]

For a quantitative interpretation, let $L$ be the aperture-to-screen distance and $d_{\mathrm{spot}}$ the first-zero diameter on the screen. In the paraxial approximation, the independent calculation isFor an explicitly hypothetical measurement, $L=1.80\mathrm{m}$ and $d_{\mathrm{spot}}=5.60\mathrm{mm}$ at $632.8\mathrm{nm}$ would imply $D\simeq 0.496\mathrm{mm}$. The scale $D^2/\lambda \simeq 0.389\mathrm{m}$ is smaller than $L$, but only by a factor of about $4.6$; this alone gives limited confidence in a high-precision Fraunhofer diffraction approximation. It is not a reported MIT measurement.

Additional recommended controls are to keep wavelength fixed, record $L$, centre and uniformly illuminate the aperture, and repeat with another $D$. Compare $d_{\mathrm{spot}}/L$, which should vary inversely with $D$. A camera readout should include blocked-beam background and avoid saturation. A scaled repetition at larger $L$ tests residual Fresnel diffraction; nonuniform illumination, optical aberrations, detector sampling and uncertainty in locating a dark ring limit the inferred $D$. The screen pattern's apparent bright edge depends on display brightness and is not a substitute for the first-zero boundary.

The Euler spiral is the planar curve parametrized by the Fresnel integrals. In the optical convention,The tangent has unit length and argument $\pi u^2/2$, so its signed curvature is $\pi u$, proportional to signed arc length. In Fresnel diffraction, an interval integral is the directed chord joining its endpoint parameters on the Cornu spiral. Chord length gives the interval factor's modulus, while chord direction gives its complex argument.

Fresnel diffraction describes diffraction when the quadratic propagation wave phase across an aperture must be retained. Its Fresnel propagator becomes the Fraunhofer diffraction description when that curvature is negligible.

A Fresnel diffraction pattern is a map of how propagation changes the wave interference of light from an aperture. It can resemble the opening, develop bright and dark structure near its edges, and eventually approach a Fraunhofer diffraction pattern. The controlling comparison involves wavelength, opening size and propagation distance together; a distance large compared with the opening need not give the far field.

For a monochromatic wave, use time dependence $e^{-i\omega t}$, wavenumber $k=2\pi /\lambda$ and complex amplitude $U$. Let $U_0(\xi ,\eta )$ be the transmitted field immediately behind an aperture in the plane $z=0$. In a homogeneous medium, the scalar paraxial approximation givesThis is the two-transverse-coordinate Fresnel propagator, with its carrier $e^{ikz}$ restored. It adds complex amplitudes before taking their squared complex modulus to obtain light intensity. It is a scalar paraxial approximation, rather than a full vector solution of Maxwell equations.^[ref]

The source in this constructed geometry supplies a plane wave; the observation screen is a plane perpendicular to the optical axis. The right panel isolates the aperture-dependent wave phase that distinguishes Fresnel diffraction from Fraunhofer diffraction. Its vertical scale is in radians, and the drawing's lengths are schematic.

Keeping this wave phase is useful when transporting a field between nearby optical components: both the shape and the complex amplitude matter for subsequent wave interference. A particularly striking feature is that propagation can create a dark centre even from a uniformly illuminated symmetric aperture; the incident wave amplitudes need not be unequal for their outgoing contributions to cancel.

For a circular aperture of radius $a$, paraxial plane-wave illumination and propagation distance $z$, define $N_F=a^2/(\lambda z)$. On axis, the radial Fresnel diffraction integral normalized by the unobstructed field givesFor a finite axial point source a distance $s$ before the aperture and detector a distance $r$ after it, replace $z$ by $sr/(s+r)$ in the phase. The formula assumes illumination amplitudes and obliquity factors approximately constant across the aperture. Odd and even numbers of Fresnel half-period zones give bright and dark axial extrema; large local intensity is compatible with global conservation of energy.

A scalar aperture approximation that sums the secondary waves reaching an observation point. For a point source with incident spherical amplitude $A{e}^{iks}/s$, one common forward-normal convention isHere $s$ and $r$ are source-to-aperture and aperture-to-observer distances and $K$ is an obliquity factor. The Fresnel diffraction approximation retains quadratic path phase; Fraunhofer diffraction additionally neglects the quadratic phase across the aperture. Kirchhoff boundary assumptions do not give an exact electromagnetic solution at a sharp edge.

A Fresnel number compares squared aperture size with the product of wavelength and propagation distance. For an optical slit of full width $D=2a$, explicitly distinguish the full-width and half-width conventions:The full-width convention is used throughout the Fresnel slit propagation example below. The definition $D^2/(\lambda z)$ also appears in the Delft treatment.^[ref]

For uniform incident wave phase, $N_D\ll 1$ makes the omitted aperture wave phase small and gives the Fraunhofer diffraction regime. When $N_D$ is of order one or larger, retain it and use Fresnel diffraction, provided the paraxial approximation still holds. These are continuous changes in accuracy, rather than a physical boundary at one selected number. For a rectangular aperture, check both transverse dimensions. Incident wavefront curvature contributes its own wave phase through $U_0$; the aperture-to-screen Fresnel number alone does not describe a spherical source's illumination.

The scaling is revealing: doubling the optical slit width at fixed wavelength and distance multiplies the Fresnel number by four. Maintaining the same Fresnel number then requires four times the distance. Two systems can consequently have the same dimensionless pattern despite very different physical lengths.

Consider a uniform plane wave of unit complex amplitude transmitted by an ideal optical slit of full width $D$. Treat the field as invariant along the optical slit height: this is one-dimensional transverse Fresnel propagation, so the prefactor is $1/\sqrt{i\lambda z}$, rather than the two-dimensional prefactor $1/(i\lambda z)$. Removing the carrier $e^{ikz}$ givesHere $I_{\mathrm{inc}}$ is incident irradiance. This infinitely long slit model describes a transverse cut away from the ends of a sufficiently long physical slit; its finite conserved quantity is transmitted power per unit optical slit height.

Put $t=\xi /D$ and $q=xD/(\lambda z)$. Substitution reduces the integral toThis form separates the pattern's shape from its physical scale. Reflection $t\mapsto -t$ proves $A(-q,N_D)=A(q,N_D)$, so the light intensity is an even function of $x$. At $N_D=0$, direct integration gives $A(q,0)=\sin (\pi q)/(\pi q)$, with limit one at $q=0$. Hence the Fraunhofer diffraction comparison is the squared sinc function. In this paragraph the unnormalized sinc function takes argument $\pi q$.

For an original example, choose wavelength $\lambda =633\mathrm{nm}$ and full width $D=0.400\mathrm{mm}$. At $z=0.0200\mathrm{m}$, $N_D\simeq 12.6382$, $N_a\simeq 3.15956$ and the edge-to-centre wave phase change is $9.92604$ radians. At $z=2.00\mathrm{m}$, $N_D\simeq 0.126382$, $N_a\simeq 0.0315956$ and that wave phase change is only $0.0992604$ radians. Even the nearer screen is fifty slit widths away; that geometric comparison alone misses its strong Fresnel diffraction.

These are numerical predictions from the displayed scalar integral, using the composite trapezoidal rule, rather than measured profiles. Each curve is divided by its own peak light intensity; the two horizontal scales differ. At 2 cm the computed central irradiance is about $1.14812I_{\mathrm{inc}}$, with brighter off-axis maxima. At 2 m it is about $0.126272I_{\mathrm{inc}}$ and is the peak. The dashed Fraunhofer diffraction curve has its own central normalization; its first zeros are at $x=\pm \lambda z/D=\pm 3.165\mathrm{mm}$. The finite-distance Fresnel diffraction curve remains slightly nonzero there.

Normalization hides the brightness change, but not the change in shape. The Fourier transform propagation multiplier has unit complex modulus, so Plancherel theorem givesThis is conservation of transmitted power per unit height in the ideal scalar model. An integral over only the displayed screen interval misses the tails; a peak-normalized profile is not a unit-integral probability density. Spreading and decreasing brightness together preserve the transmitted power.

A family of integral transforms associated with a two-by-two real matrix of determinant one. For $B\ne 0$, one optical normalization has kernel proportional to $\exp [ik(A{x}^2+D{x}^{\prime 2}-2x{x}^{\prime })/(2B)]$. It includes free-space Fresnel diffraction, quadratic phase multiplication and the Fourier transform. The optical amplitude normalization in one transverse dimension is $1/\sqrt{i\lambda B}$ with a consistent phase branch.

A documented experiment uses curved illumination rather than the uniform Fresnel slit propagation model. Laboratory Exercise No. 4: Diffraction, MIT 6.237, Fall 2024, printed pp. 5–6, gives a helium–neon laser at $632.8\mathrm{nm}$, a microscope objective optical lens, a circular aperture and a screen 5–40 cm from the object. Preparation includes comparing central brightness in Fresnel diffraction and Fraunhofer diffraction. A 40× objective produces a nonplanar incident wavefront; the opening is placed less than 1 cm from it. With objective-to-screen distance fixed, students move the opening, sketch patterns, and record positions of consecutive central nulls to estimate aperture size. The Fraunhofer diffraction pattern supplies a regime comparison. A separate spherical-obstacle task observes a bright centre inside the shadow. The handout supplies procedures, rather than measured distances or profiles.^[ref]

The circular-opening test probes cancellation of complex amplitudes at the centre while the source wavefront curvature changes. To interpret it, include that curvature in the aperture field $U_0$ and use a circular aperture, rather than substituting the plane-illuminated Fresnel slit propagation profile. Useful additional controls are to retain the same laser and opening, centre the illumination, repeat position readings and check background with the beam blocked. If a camera replaces the visual screen readout, avoid saturation and calibrate its position scale. Alignment, imperfect aperture edges, illumination variation and limited contrast near a null constrain the inferred size. A bright central obstacle spot illustrates why the sharp shadows of geometrical optics can fail; it does not imply that every Fresnel diffraction pattern has a bright centre.

A high-pressure Mercury arc lamp is a conventional source for proximity photolithography. An electric discharge excites Mercury vapour; radiative transitions produce strong spectral lines, selected by filters and delivered through collimating optics. Common lines are $365\mathrm{nm}$ (i-line) and $436\mathrm{nm}$ (g-line). The finite mask-wafer gap gives near-field Fresnel diffraction, rather than the far-field approximation. Diffracted light spreads into nominally dark regions, limiting the minimum printable feature.

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.

Fresnel diffraction retains the quadratic transverse propagation phase; Fraunhofer diffraction neglects it and leaves an aperture Fourier transform. Expanding the propagation distance gives $z+y^2/(2z)+\cdots$. The aperture phase variation is of order $\pi {\rho }^2/(\lambda z)$, so the crossover scale is $z\sim {\rho }^2/\lambda$, with far-field propagation requiring a much smaller Fresnel number than1.

Here $\lambda \simeq 0.0100\mathrm{m}$ and the full gap width gives ${\rho }^2/\lambda =25\mathrm{m}$, comparable with the16m range. More decisively, measured from the grazing line of sight to the far gap edge, the quadratic phase reaches $\pi (0.5{)}^2/(0.01\times 16)=4.91\mathrm{rad}$. Fresnel diffraction is required; conventions using a half-width change the quoted Fresnel number but not this phase test.

Define $F(w)=C(w)+iS(w)$ with the normalization in the paper. The grazing-edge gap spans $w=0$ toThe Cornu chord power normalization uses a received field proportional to $F(w_g)-F(0)$, while no fence gives $F(\infty )-F(-\infty )=1+i$. Numerical Fresnel integration gives $C(w_g)=0.3245$, $S(w_g)=0.4818$, and thereforeThe two labelled child answers specify the corresponding vectors.

For the largest contiguous Fresnel aperture, a freely chosen single contiguous opening must maximize the chord between two points of the Cornu spiral. Its longest chord joins $F(-w_{\ast })$ to $F(w_{\ast })$, where the first maximum radius satisfies $C(w_{\ast })\cos (\pi w_{\ast }^2/2)+S(w_{\ast })\sin (\pi w_{\ast }^2/2)=0$. This gives $w_{\ast }=1.2094$ andHere x is signed displacement along the fence from the grazing optical axis. The pointwise Fresnel gain versus integrated transmitted power distinction explains why an aperture can concentrate more power at one receiver than the fully open field through constructive interference; this is not an increase of integrated transmitted power.

The existing-opening constraint in Fresnel aperture optimization matters literally: the original gap already occupies $[0,0.5]\mathrm{m}$. The symmetric optimum would require filling its far end. If only further removal is permitted and the final opening must be connected, maximize the chord subject to $w_{-}\le 0$ and $w_{+}\ge 1.768$. The constrained optimum is approximatelyThese are different optimization assumptions; both are stated so the original gap is not silently replaced. The model treats the fence as an ideal opaque screen and ignores ground reflections or additional propagation paths. ^[ref]

The Arago spot is a bright axial point inside the geometrical shadow of a circular opaque object. Under coherent illumination, contributions from points around its circular edge have equal optical path to an axial observation point and interfere constructively. It is a Fresnel diffraction effect, so geometrical rays alone miss it. In the ideal paraxial plane-wave model, Babinet's principle gives $U_{\mathrm{disk}}=U_0-U_{\mathrm{aperture}}=U_0{e}^{ik{a}^2/(2z)}$ on axis for disk radius $a$; thus the axial intensity equals the unobstructed intensity in that model. The whole shadow is not bright: off-axis phases differ and form a localized central feature. Finite source size, poor coherence or an imperfect edge reduce its visibility.

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.

The original diagram puts the upper edge of the obstruction exactly on the straight line from source to detector. The detector is therefore at the geometric shadow boundary, not deep inside the shadow. In the optical convention, the Cornu spiral iswith limiting points $(-1/2,-1/2)$ and $(1/2,1/2)$. The unobstructed Fresnel diffraction amplitude is the chord joining both endpoints, proportional to $1+i$. The open half-plane supplies only the chord from $(0,0)$ to $(1/2,1/2)$, proportional to $(1+i)/2$. The coordinate perpendicular to the page contributes the same factor in both cases.This is the half-plane shadow-boundary intensity. The missing half of the aperture removes half the field here, so taking half the intensity would be incorrect.

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.

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.

The expression is a scalar diffraction theory integral for a monochromatic wave. The complex quantity ${\psi }_P$ represents a field complex amplitude, with physical field obtained by taking the real part after multiplication by $e^{-i\omega t}$. The light intensity is proportional to $|{\psi }_P{|}^2$. The coordinates $x,y$ label transverse positions in the aperture plane, $\Sigma$ is its transmitting region, and $h(x,y)$ is the incident field multiplied by the aperture's complex transmission. The wavenumber is $k=2\pi /\lambda$ in the propagation medium, and $R$ is the axial propagation distance if $h$ already includes the incident field.

To derive the Fresnel diffraction, start from Huygens-Fresnel principle contributions with propagation factor $e^{iks}/s$. With the coordinate origin at the projection of $P$ into the aperture plane,The paraxial approximation requires $\rho /R\ll 1$, where ${\rho }^2=x^2+y^2$. Replace the slowly varying $1/s$ and the obliquity factor by their axial values, but retain the quadratic variation of wave phase. Neglect of the next wave phase term requires $k{\rho }^4/(8R^3)\ll 1$ over the relevant aperture. Thus, apart from a common constant,The quadratic wave phase may be of order one or much larger; setting it to zero would require the stronger conditions for Fraunhofer diffraction. The scalar treatment also neglects the detailed vector field at aperture edges and assumes dimensions large enough compared with wavelength for this approximation to be useful.

For a finite-distance axial point source at distance $p$ and an observer at distance $q$, the incident field contributes its own quadratic wave phase. If $h$ instead denotes only the aperture transmission, the combined curvature is described by an effective distanceThese are two equivalent conventions for distributing the incident wave phase between $h$ and the exponential. Both distances must be included in the later calculation.

Write $p$ and $q$ for the axial source and observation distances, and put $R_{\mathrm{eff}}=pq/(p+q)$. The aperture is $-d/2<x,y<d/2$. Under the Fresnel diffraction, the spherical incident wave amplitude is effectively constant over this small aperture, while its quadratic wave phase must be retained. Therefore the two transverse integrals separate:where $K$ contains the source strength, the propagation distances and a common phase factor.

Use the dimensionless variable $u=x\sqrt{2/(\lambda R_{\mathrm{eff}})}$ and define the optical normalization of Fresnel integralsThe oddness of $F$ makes the bracket $\sqrt{2\lambda R_{\mathrm{eff}}}F(w)$, soThis square-aperture Fresnel diffraction normalization is especially useful for comparing with no aperture. In the same Fresnel diffraction, the unrestricted one-dimensional Fresnel integral is $\sqrt{\lambda R_{\mathrm{eff}}}{e}^{i\pi /4}$; its square is $i\lambda R_{\mathrm{eff}}$. Thus, writing ${\psi }_0$ for the unobscured field at the same point,The normalization has the correct no-aperture limit because $C(\infty )=S(\infty )=1/2$ in this convention.