# Fraunhofer diffraction
**Fraunhofer diffraction** is the diffraction pattern formed by light propagating to a observation region far from an aperture, where the arriving wavefronts can be treated as plane waves. It is the far-field limit of scalar diffraction theory, contrasted with [Fresnel diffraction](https://www.edgepedia.org/fresnel-diffraction), the near-field regime in which wavefront curvature still matters. Joseph von Fraunhofer first proposed the analysis in 1823.[1](https://ufdcimages.uflib.ufl.edu/UF/E0/01/34/24/00001/chung_h.pdf) The regime matters practically because it governs the diffraction patterns of slits, gratings, and telescope apertures, and thereby sets the resolution limit of imaging systems.[2](https://farside.ph.utexas.edu/teaching/315/Waves/node98.html)
| Key fact | Detail |
|---|---|
| Defining approximation | Source and observation points far enough from the aperture that quadratic phase terms can be neglected; waves are treated as plane waves[3](https://www.physics.utoronto.ca/~phy485/LaserPhysics/dloads/ModOptNotes2010.pdf) |
| Central result | The far-field amplitude pattern is the Fourier transform of the field distribution at the aperture, evaluated at spatial frequencies \( x/(\lambda z) \), \( y/(\lambda z) \)[3](https://www.physics.utoronto.ca/~phy485/LaserPhysics/dloads/ModOptNotes2010.pdf) • [4](https://interactivetextbooks.tudelft.nl/tn2421/content/Chap6_Diffraction/DiffractiveOptics_2022_01Clean.html) |
| Distance scale | For a He-Ne laser (\( \lambda = 6 \times 10^{-7} \) m) and a 1 mm aperture, the far field begins at roughly \( z \gtrsim 3 \) m[3](https://www.physics.utoronto.ca/~phy485/LaserPhysics/dloads/ModOptNotes2010.pdf) |
| Single slit | Minima at \( a \sin\theta = m\lambda \), \( m = 1, 2, 3, \ldots \); intensity \( I = I_0 \, \mathrm{sinc}^2 \alpha \) with \( \alpha = (\pi a/\lambda)\sin\theta \)[5](https://www.physics.purdue.edu/~jones105/phys42200_Spring2013/notes/Phys42200_Lecture38.pdf) |
| Circular aperture | Airy disk pattern; angular radius of the first dark ring \( \delta\theta = 1.22\,\lambda/D \); the central disk holds 0.86 of the pattern's total energy[3](https://www.physics.utoronto.ca/~phy485/LaserPhysics/dloads/ModOptNotes2010.pdf) • [2](https://farside.ph.utexas.edu/teaching/315/Waves/node98.html) |
| Resolution limit | Rayleigh criterion: two point sources are resolvable when their angular separation exceeds \( 1.22\,\lambda/D \)[2](https://farside.ph.utexas.edu/teaching/315/Waves/node98.html) |
| Lens shortcut | A positive lens produces the Fraunhofer pattern of an incident beam in its back focal plane, regardless of focal length[3](https://www.physics.utoronto.ca/~phy485/LaserPhysics/dloads/ModOptNotes2010.pdf) |
## The Fraunhofer approximation
Diffraction from an aperture is calculated from the [Huygens–Fresnel principle](https://www.edgepedia.org/huygens-fresnel-principle), which treats every point in the aperture as a source of secondary wavelets whose interference forms the observed pattern.[6](https://users.physics.ox.ac.uk/~lvovsky/471/labs/fraunhofer.pdf) The exact treatment sums spherical waves. The Fresnel (paraxial) approximation replaces these with parabolic waves, which introduces a quadratic phase factor \( \exp\left[ik(x'^2+y'^2)/(2z)\right] \) in the diffraction integral, where \( z \) is the propagation distance and \( k = 2\pi/\lambda \).[4](https://interactivetextbooks.tudelft.nl/tn2421/content/Chap6_Diffraction/DiffractiveOptics_2022_01Clean.html)
The Fraunhofer approximation makes one further step: it drops that quadratic phase factor entirely, which is justified when the source and observation points are far from the aperture.[3](https://www.physics.utoronto.ca/~phy485/LaserPhysics/dloads/ModOptNotes2010.pdf) • [4](https://interactivetextbooks.tudelft.nl/tn2421/content/Chap6_Diffraction/DiffractiveOptics_2022_01Clean.html) The remaining integral is a pure [Fourier transform](https://www.edgepedia.org/fourier-transform) of the aperture field. The approximation is sufficient but not necessary; phase cancellation in the neglected higher-order terms means the far-field pattern can sometimes be observed somewhat closer than the nominal condition requires.[3](https://www.physics.utoronto.ca/~phy485/LaserPhysics/dloads/ModOptNotes2010.pdf)
## The Fourier transform relation
[Fourier optics](https://www.edgepedia.org/fourier-optics)' key result states that the Fourier transform of the field distribution across the aperture gives the diffraction pattern observed in the far field, known as the Fraunhofer pattern.[3](https://www.physics.utoronto.ca/~phy485/LaserPhysics/dloads/ModOptNotes2010.pdf) In the Delft formulation,
\[ U(x,y,z) \approx \frac{e^{ikz} e^{\frac{ik(x^2+y^2)}{2z}}}{i\lambda z}\, \mathcal{F}(U_0)\left(\frac{x}{\lambda z}, \frac{y}{\lambda z}\right), \]
where \( U_0 \) is the field in the aperture plane.[4](https://interactivetextbooks.tudelft.nl/tn2421/content/Chap6_Diffraction/DiffractiveOptics_2022_01Clean.html) The transform is evaluated at spatial frequencies \( x/(\lambda z) \) and \( y/(\lambda z) \), which scale as \( 1/z \), and the overall field is proportional to \( 1/z \). Once the Fraunhofer approximation is accurate, increasing the distance only widens the pattern and lowers its amplitude; the shape no longer changes, and total energy is preserved.[4](https://interactivetextbooks.tudelft.nl/tn2421/content/Chap6_Diffraction/DiffractiveOptics_2022_01Clean.html) This is a sharp contrast with the Fresnel region, where the pattern's shape changes substantially with distance, evolving from roughly a copy of the aperture into the Fourier-transform pattern.[4](https://interactivetextbooks.tudelft.nl/tn2421/content/Chap6_Diffraction/DiffractiveOptics_2022_01Clean.html)
**Practical consequence.** Because the far-field pattern depends only on observation direction and not on distance, a lens can replace a long propagation path: a positive lens performs a Fourier transform of the input beam, so its focal plane shows the Fraunhofer pattern of whatever field is placed in front of it.[3](https://www.physics.utoronto.ca/~phy485/LaserPhysics/dloads/ModOptNotes2010.pdf) This is the basis of coherent optical processing, in which a system's transfer function is the pupil function evaluated at \( -\lambda z \) times the spatial frequency coordinates.[7](https://opticsthewebsite.com/Rectangular)
## Standard aperture patterns
**Single slit.** For a slit of width \( a \) under plane-wave illumination, the intensity is \( I_\theta = I_m [(\sin\alpha)/\alpha]^2 \) with \( \alpha = (\pi a/\lambda)\sin\theta \); minima fall at \( a\sin\theta = m\lambda \).[5](https://www.physics.purdue.edu/~jones105/phys42200_Spring2013/notes/Phys42200_Lecture38.pdf) • [8](https://web.physics.ucsb.edu/~lecturedemonstrations/Composer/Pages/84.06.html) The first zero moves closer to the center as the slit widens, an inverse dependence of pattern width on aperture size.[9](https://mysite.du.edu/~etuttle/optics/ywave.htm)
**Rectangular aperture.** A uniformly illuminated \( \ell_x \times \ell_y \) rectangle gives a product of two sinc functions, \( I \propto \mathrm{sinc}^2(\ell_x x/\lambda_0 z)\,\mathrm{sinc}^2(\ell_y y/\lambda_0 z) \), with the central peak extending \( \Delta x = 2\lambda_0 z/\ell_x \) between the first nodes.[3](https://www.physics.utoronto.ca/~phy485/LaserPhysics/dloads/ModOptNotes2010.pdf)
**Circular aperture.** A uniformly illuminated circular aperture of radius \( R \) yields the Airy pattern, \( I(\theta) \propto [2J_1(ka\theta)/(ka\theta)]^2 \), a central disk surrounded by much fainter concentric rings.[2](https://farside.ph.utexas.edu/teaching/315/Waves/node98.html) The angular radius of the [Airy disk](https://www.edgepedia.org/airy-disk) is \( \delta\theta = 1.22\,\lambda/D \), where \( D = 2a \) is the aperture diameter; the factor 1.22 comes from the first zero of the [Bessel function](https://www.edgepedia.org/bessel-function) \( J_1 \) at \( z = 3.83171 \).[2](https://farside.ph.utexas.edu/teaching/315/Waves/node98.html) The central disk contains 0.86 of the total energy in the pattern.[3](https://www.physics.utoronto.ca/~phy485/LaserPhysics/dloads/ModOptNotes2010.pdf)
**Slits and gratings.** For two slits of width \( a \) separated by \( b \), the Fraunhofer intensity is \( I(\theta) = 4I_0 (\sin\beta/\beta)^2 \cos^2\alpha \): the single-slit sinc² envelope modulates the finer two-slit interference fringes, whose maxima satisfy \( b\sin\theta = m\lambda \).[5](https://www.physics.purdue.edu/~jones105/phys42200_Spring2013/notes/Phys42200_Lecture38.pdf) • [6](https://users.physics.ox.ac.uk/~lvovsky/471/labs/fraunhofer.pdf) Adding more equidistant slits keeps the maxima at the same angular positions but sharpens them; the limiting case is the diffraction grating, whose maxima obey \( d\sin\theta = m\lambda \) and which is widely used in monochromators and spectrometers.[6](https://users.physics.ox.ac.uk/~lvovsky/471/labs/fraunhofer.pdf)
## Distinguishing Fraunhofer from Fresnel diffraction
The two regimes can be separated quantitatively by counting Fresnel zones, the annular regions of an expanding wavefront that reach the observation point in half-wavelength steps. A criterion based on a Young double-pinhole experiment shows that Fraunhofer diffraction is obtained when the aperture subtends only the first [Fresnel zone](https://www.edgepedia.org/fresnel-zone) or a fraction of it from the observation point, while Fresnel diffraction applies when more than one zone is subtended; because the test is interferometric, it applies to apertures of any shape.[10](https://www.sciencedirect.com/science/article/pii/S0030401801016054) In the equivalent Fresnel-number picture, the distance to the mask increases as the [Fresnel number](https://www.edgepedia.org/fresnel-number) \( N_F \) decreases, moving the observed pattern from a near-field copy of the aperture to the Fraunhofer Fourier-transform pattern.[4](https://interactivetextbooks.tudelft.nl/tn2421/content/Chap6_Diffraction/DiffractiveOptics_2022_01Clean.html)
At optical wavelengths the direct far field is demanding: with \( \lambda_0 = 6 \times 10^{-7} \) m and a 1 mm aperture, the Toronto notes give \( z \gtrsim 3 \) m as the condition for observing the Fraunhofer pattern.[3](https://www.physics.utoronto.ca/~phy485/LaserPhysics/dloads/ModOptNotes2010.pdf) Lecture demonstrations exploit a diverging lens to show Fresnel patterns, Fraunhofer patterns, and the dynamic transition between them within a few meters.[11](https://arxiv.org/abs/0803.0120)
## Role in imaging and astronomy
When a point source such as a star is observed through a telescope, the image seen is the Fraunhofer diffraction pattern of the objective aperture: an Airy disk with faint rings.[2](https://farside.ph.utexas.edu/teaching/315/Waves/node98.html) The Rayleigh criterion, named for John William Strutt, the 3rd Baron Rayleigh, states that a pair of sources is resolvable only if the central peak of one diffraction pattern lies no closer than the radius of the other's Airy disk, giving \( \Delta\theta > 1.22\,\lambda/D \).[2](https://farside.ph.utexas.edu/teaching/315/Waves/node98.html) • [12](https://astronomy.swinburne.edu.au/cosmos/R/Rayleigh%2BCriterion) For Hubble's 2.4 m aperture at 500 nm this is about \( 2 \times 10^{-7} \) rad, or 0.05 arcsec, while Spitzer's 85 cm aperture at 24 microns gives roughly \( 3 \times 10^{-5} \) rad, about 7.1 arcsec, illustrating how longer wavelengths and smaller apertures enlarge the diffraction spot.[13](https://coolwiki.ipac.caltech.edu/index.php/Diffraction) In radio and submillimeter astronomy the same theory links the illumination of a telescope aperture to the far-field beam pattern, and the Fourier transform of the power pattern gives the telescope's transfer function.[14](https://www.submm.caltech.edu/~cdd/SPIRE/20081027_diffraction/)
## References
1. [Chung, H. Fraunhofer Diffraction, §2.2.2, University of Florida thesis](https://ufdcimages.uflib.ufl.edu/UF/E0/01/34/24/00001/chung_h.pdf)
2. [Two-Dimensional Fourier Optics, University of Texas at Austin lecture notes](https://farside.ph.utexas.edu/teaching/315/Waves/node98.html)
3. [Fraunhofer Diffraction, Fourier Optics and Holography, University of Toronto modified optics notes (2010)](https://www.physics.utoronto.ca/~phy485/LaserPhysics/dloads/ModOptNotes2010.pdf)
4. [Scalar Diffraction Optics, TU Delft Interactive Optics textbook (2022)](https://interactivetextbooks.tudelft.nl/tn2421/content/Chap6_Diffraction/DiffractiveOptics_2022_01Clean.html)
5. [Phys 42200 Lecture 38, Purdue University](https://www.physics.purdue.edu/~jones105/phys42200_Spring2013/notes/Phys42200_Lecture38.pdf)
6. [Fraunhofer Diffraction laboratory guide, University of Oxford](https://users.physics.ox.ac.uk/~lvovsky/471/labs/fraunhofer.pdf)
7. [Rectangular Aperture Diffraction, Optics: The Website](https://opticsthewebsite.com/Rectangular)
8. [Laser beam diffracted through various slits, UC Santa Barbara lecture demonstrations](https://web.physics.ucsb.edu/~lecturedemonstrations/Composer/Pages/84.06.html)
9. [Interference and Diffraction in Optics, University of Denver](https://mysite.du.edu/~etuttle/optics/ywave.htm)
10. [Medina, F. F. Distinguishing between Fraunhofer and Fresnel diffraction by the Young's experiment, Optics Communications 200 (2001) 15–22](https://www.sciencedirect.com/science/article/pii/S0030401801016054)
11. [Dynamic transition between Fresnel and Fraunhofer diffraction patterns – a lecture experiment, arXiv:0803.0120](https://arxiv.org/abs/0803.0120)
12. [Rayleigh Criterion, COSMOS, Swinburne University of Technology](https://astronomy.swinburne.edu.au/cosmos/R/Rayleigh%2BCriterion)
13. [Diffraction, CoolWiki, IPAC Caltech](https://coolwiki.ipac.caltech.edu/index.php/Diffraction)
14. [Diffraction and Filtering, Caltech Submillimeter Observatory SPIRE notes](https://www.submm.caltech.edu/~cdd/SPIRE/20081027_diffraction/)
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*Topic: Encyclopedia › Physical world and mathematics › Physics › Classical physics › Waves and optics › Physical and wave optics › Interference and diffraction › Diffraction theory*
*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*
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