Fraunhofer diffraction

Fraunhofer diffraction is the far-field approximation for diffraction: contributions from an aperture reach a selected observation direction with approximately linear path-dependent wave phases. Adding their complex amplitudes gives a Fourier transform of the field across the aperture. It also describes the appropriate rear focal plane of an ideal optical lens under coherent plane-wave illumination. The diffraction grating examples use this regime. ^[ref]; ^[ref].

For a slit of width $a$, wavelength $\lambda$ and screen distance $L$, sufficient conditions near the forward direction areThe last two statements are equivalent: the important comparison is with $a^2/\lambda$, rather than just with the slit width. For a finite rectangular aperture, check the corresponding condition for both transverse dimensions. The single-slit diffraction calculation treats the transverse cut associated with width $a$. ^[ref].

A short explanation is that the distance from slit coordinate $x$ to an observation point at distance $R$ and angle $\theta$ isThe linear term supplies the phase difference; the quadratic term supplies a wave phase error of order $a^2/(\lambda R)$, since the wavenumber is $2\pi /\lambda$. Near the forward direction $R\approx L$, so the stated condition makes this error small. This is why a distant screen permits the simple single-slit diffraction formula. ^[ref].

The far-field approximation neglects quadratic variation of propagation phase across an aperture of characteristic size a at observation distance D. For two narrow slits separated by s, their path difference remains $s\sin \theta$ at a finite observation angle. The additional substitution $\sin \theta \simeq Y/D$ requires a small observation angle; it does not follow from the Fraunhofer approximation itself.

A diffraction envelope is the angular factor imposed by the shape and width of each opening in an interference pattern. For two equally illuminated rectangular slits of width $b$ and separation $s$, Fraunhofer diffraction gives $I(\theta )=4I_s{\mathrm{sinc}}^2(\pi b\sin \theta /\lambda ){\cos }^2(\pi s\sin \theta /\lambda )$, where $\mathrm{sinc}u=\sin u/u$ and $I_s$ is one slit's on-axis light intensity. The first factor limits the observable angular range even when the rapid interference fringes remain perfectly coherent.

The relevant aperture size $b$ spans the entire coherent array, including the separated elements, rather than one element or a single baseline increment. A quadratic propagation-phase error is small only when $b^2/(\lambda R)\ll 1$ under the paraxial approximation. The often-used boundary $2b^2/\lambda$ is a convention for an allowed phase error, not a sharp physical transition. Individual-element patterns still supply an angular envelope in the far field.

Let a coherent array of identical narrow slits have total far-field complex amplitude zero at a chosen direction. Reversing one slit's phase by $\pi$ changes its amplitude from $a_j$ to $-a_j$, so the new total is $-2a_j$. Its intensity is four times the single-slit intensity, independent of which slit was changed. For $N$ equally spaced slits, adjacent phase step $2\pi m/N$ gives such a zero when $m$ is not a multiple of $N$. A finite slit envelope multiplies the common intensity factor.

The maxima of the full two-slit intensity need not lie at maxima of its cosine-squared interference factor, because the single-slit envelope has a nonzero slope. Differentiating its logarithm gives $[\pi w/(\lambda L)](\cot u-1/u)=[\pi d/(\lambda L)]\tan v$, where $u=\pi wX/(\lambda L)$ and $v=\pi dX/(\lambda L)$. Solving this equation in the appropriate fringe gives the true maximum. The usual fringe-center estimate is useful when the envelope varies slowly across a fringe.

Reversing the phase on one half of an otherwise uniform slit makes its aperture transmission odd. Its Fourier amplitude is $4{\sin }^2(qb/4)/(iq)$, where $q=k\sin \theta$, so the axis is dark despite the removable denominator singularity. The largest symmetric lobes satisfy $\tan s=2s$, with first positive root approximately $1.16556$. Doubling an ideal half-wave layer restores equal phases and the ordinary bright-centred single-slit pattern. Total transmitted power is conserved if the layer has no absorption or reflection.

A uniformly illuminated rectangle of width $w$ and height $h$ has field proportional to $\mathrm{sinc}(q_{x}w/2)\mathrm{sinc}(q_{y}h/2)$, with angular spatial wavenumbers $q_x,q_y$ measured in inverse length. The first envelope zeros give $w=2\pi /|q_{x,0}|$ and $h=2\pi /|q_{y,0}|$. Two identical rectangles with horizontal centre separation $s$ introduce an additional $\cos (q_{x}s/2)$ factor, so the interference period is $\Delta q_x=2\pi /s$.

The factors $2\pi$ depend on using angular spatial frequency; axes in cycles per length would use the reciprocal widths directly. Fourier light intensity alone cannot determine an overall aperture translation or distinguish a mask from its inverted copy. Extra assumptions about shape and coherent illumination are necessary for a unique geometrical reconstruction.

For four identical long slits of width $a$ centred at $x=\pm D/2,\pm 3D/2$, the aperture Fourier transform givesHere $\mathrm{sinc}x=\sin x/x$. Normalized to the on-axis light intensity,Principal array directions have $qD=2\pi m$; a single-slit diffraction envelope zero cancels such an order when $ma/D$ is a nonzero integer. The closest central array zeros satisfy $qD=\pm \pi /2$. Finite slit width generally shifts the actual maxima of the complete product slightly away from the nominal principal array directions.

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.

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.

For mutually incoherent wavelengths, intensities add rather than amplitudes: $I(\theta )=\int S(\lambda ){\mathrm{sinc}}^2[\pi a\sin \theta /\lambda ]d\lambda$, where $\mathrm{sinc}q=\sin q/q$ and S includes source strength and detector response. All wavelengths share the central maximum but their minima occur at different angles. A smooth white spectrum therefore has a white centre, wavelength-dependent edge colours, no common first dark minimum and increasingly overlapped, low-contrast outer fringes. A particular hue needs the actual spectrum and colour response.

A thin lens placed immediately after an aperture can reproduce its free-propagated pattern from distance L on a nearer screen d if $1/f=1/d-1/L$. In the paraxial Fresnel integral, the lens changes the aperture's quadratic propagation phase from $k{x}^2/(2d)$ to $k{x}^2(1/d-1/f)/2=k{x}^2/(2L)$. The cross term $-kxy/d$ then matches free propagation at the old-screen coordinate $Y=Ly/d$. Intensities have the same shape under transverse scale d/L, with an overall brightness factor in a lossless lens of sufficient aperture. For a narrow uniformly illuminated slit, the far-field first-zero estimate on the new screen is $d\lambda /a$.