Kirchhoff's diffraction formula^[1][2] (also calledFresnel–Kirchhoff diffraction formula) approximates light intensity and phase in opticaldiffraction: light fields in the boundary regions of shadows. The approximationcan be used to model lightpropagation in a wide range of configurations, eitheranalytically or usingnumerical modelling. It gives an expression for the wave disturbance when amonochromaticspherical wave is the incoming wave of a situation under consideration. This formula is derived by applying theKirchhoff integral theorem, which uses theGreen's second identity to derive the solution to the homogeneousscalar wave equation, to a spherical wave with some approximations.

TheHuygens–Fresnel principle is derived by the Fresnel–Kirchhoff diffraction formula.

Kirchhoff's integral theorem, sometimes referred to as the Fresnel–Kirchhoff integral theorem,^[3] usesGreen's second identity to derive the solution of the homogeneousscalar wave equation at an arbitrary spatial positionP in terms of the solution of the wave equation and its first order derivative at all points on an arbitrary closed surface${\displaystyle S}$ as the boundary of some volume includingP.

The solution provided by the integral theorem for amonochromatic source is^[1]: 378$${\displaystyle U(\mathbf{\text{P}})=\frac{1}{4\pi }{\int }_S\left[U\frac{\mathrm{\partial }}{\mathrm{\partial }\mathbf{\text{n}}}\left(\frac{e^{iks}}{s}\right)-\frac{e^{iks}}{s}\frac{\mathrm{\partial }U}{\mathrm{\partial }\mathbf{\text{n}}}\right]dS,}$$where${\displaystyle U}$ is the spatial part of the solution of the homogeneousscalar wave equation (i.e.,${\displaystyle V(\mathbf{r},t)=U(\mathbf{r})e^{-i\omega t}}$ as the homogeneous scalar wave equation solution),k is thewavenumber, ands is the distance fromP to an (infinitesimally small) integral surface element, and${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }n}}$ denotes differentiation along the integral surface element normal unit vector${\displaystyle n}$ (i.e., anormal derivative), i.e.,${\displaystyle \frac{\mathrm{\partial }f}{\mathrm{\partial }n}=\mathrm{\nabla }f\cdot n}$. Note that the surface normal or the direction of${\displaystyle n}$ is toward the inside of the enclosed volume in this integral; if the more usualouter-pointing normal is used, the integral will have the opposite sign. And also note that, in the integral theorem shown here,${\displaystyle n}$ andParevector quantities while other terms arescalar quantities.

For the below cases, the following basic assumptions are made.

• The distance between apoint source of waves and an integral area, the distance between the integral area and an observation pointP, and the dimension of openingS are much greater than the wave wavelength${\displaystyle \lambda }$.
• ${\displaystyle U}$ and${\displaystyle \frac{\mathrm{\partial }U}{\mathrm{\partial }n}=\mathrm{\nabla }U\cdot n}$ are discontinuous at the boundaries of the aperture, calledKirchhoff's boundary conditions. This may be related with another assumption that waves on an aperture (or an open area) is same to the waves that would be present if there was no obstacle for the waves.

Consider a monochromatic point source atP₀, which illuminates an aperture in a screen. Theintensity of the wave emitted by a point source falls off as the inverse square of the distance traveled, so the amplitude falls off as the inverse of the distance. The complex amplitude of the disturbance at a distance${\displaystyle r}$ is given by

$${\displaystyle U(r)=\frac{a{e}^{ikr}}{r},}$$where${\displaystyle a}$ represents themagnitude of the disturbance at the point source.

The disturbance at a spatial positionP can be found by applying theKirchhoff's integral theorem to the closed surface formed by the intersection of a sphere of radiusR with the screen. The integration is performed over the areasA₁,A₂ andA₃, giving$${\displaystyle U(P)=\frac{1}{4\pi }\left[{\int }_{A_1}+{\int }_{A_2}+{\int }_{A_3}\right]\left(U\frac{\mathrm{\partial }}{\mathrm{\partial }n}\left(\frac{e^{iks}}{s}\right)-\frac{e^{iks}}{s}\frac{\mathrm{\partial }U}{\mathrm{\partial }n}\right)dS.}$$

To solve the equation, it is assumed that the values of${\displaystyle U}$ and${\displaystyle \frac{\mathrm{\partial }U}{\mathrm{\partial }n}}$ in the aperture areaA₁ are the same as when the screen is not present, so at the positionQ,$${\displaystyle U_{A_1}=\frac{a{e}^{ikr}}{r},}$$$${\displaystyle \frac{\mathrm{\partial }{U}_{A_1}}{\mathrm{\partial }n}=\mathrm{\nabla }{U}_{A_1}\cdot n=\frac{a{e}^{ikr}}{r}\left[ik-\frac{1}{r}\right]\cos (n,r),}$$where${\displaystyle r}$ is the length of the straight lineP₀Q, and${\displaystyle (n,r)}$ is the angle between a straightly extended version ofP₀Q and the (inward) normal to the aperture. Note that so${\displaystyle \cos (n,r)}$ is a positive real number onA₁.

AtQ, we also have$${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }n}\left(\frac{e^{iks}}{s}\right)=\frac{e^{iks}}{s}\left[ik-\frac{1}{s}\right]\cos (n,s),}$$where${\displaystyle s}$ is the length of the straight linePQ, and${\displaystyle (n,s)}$ is the angle between a straightly extended version ofPQ and the (inward) normal to the aperture. Note that so${\displaystyle \cos (n,s)}$ is a negative real number onA₁.

Two more following assumptions are made.

• In the above normal derivatives, the terms${\displaystyle \frac{1}{r}}$ and${\displaystyle \frac{1}{s}}$ in the both square brackets are assumed to be negligible compared with thewavenumber${\displaystyle k=\frac{2\pi }{\lambda }}$, means${\displaystyle r}$ and${\displaystyle s}$ are much greater than thewavelength${\displaystyle \lambda }$.
• Kirchhoff assumes that the values of${\displaystyle U}$ and${\displaystyle \frac{\mathrm{\partial }U}{\mathrm{\partial }n}}$ on the opaque areas marked byA₂ are zero. This implies that${\displaystyle U}$ and${\displaystyle \frac{\mathrm{\partial }U}{\mathrm{\partial }n}}$ are discontinuous at the edge of the apertureA₁. This is not the case, and this isone of the approximations used in deriving the Kirchhoff's diffraction formula.^[4][5] These assumptions are sometimes referred to asKirchhoff's boundary conditions.

The contribution from the hemisphereA₃ to the integral is expected to be zero, and it can be justified by one of the following reasons.

1. Make the assumption that the source starts to radiate at a particular time, and then makeR large enough, so that when the disturbance atP is being considered, no contributions fromA₃ will have arrived there.^[1] Such a wave is no longermonochromatic, since a monochromatic wave must exist at all times, but that assumption is not necessary, and a more formal argument avoiding its use has been derived.^[6]
2. A wave emanated from the apertureA₁ is expected to evolve toward a spherical wave as it propagates (Water wave examples of this can be found in many pictures showing a water wave passing through a relatively narrow opening.). So, ifR is large enough, then the integral onA₃ becomes$${\displaystyle U(P)\sim -\frac{ia}{2\lambda }{\int }_{A_3}\frac{e^{2ik(r^{\prime })}}{r^{\prime 2}}[\cos (n,r^{\prime })-\cos (n,r^{\prime })]d{A}_3=-\frac{ia}{2\lambda }{\int }_{A_3}{e}^{2ik(r^{\prime })}[\cos (n,r^{\prime })-\cos (n,r^{\prime })]d\mathrm{\Omega }=0}$$ where${\displaystyle r^{\prime }}$ and${\displaystyle d\mathrm{\Omega }}$ are the distance from the center of the apertureA₁ to an integral surface element andthe differential solid angle in the spherical coordinate system respectively.

As a result, finally, the integral above, which represents the complex amplitude atP, becomes$${\displaystyle U(P)=-\frac{ia}{2\lambda }{\int }_{A_1}\frac{e^{ik(r+s)}}{rs}[\cos (n,r)-\cos (n,s)]d{A}_1.}$$

This is theKirchhoff orFresnel–Kirchhoff diffraction formula.

TheHuygens–Fresnel principle can be derived by integrating over a different closed surface (the boundary of some volume having an observation pointP). The areaA₁ above is replaced by a part of a wavefront (emitted from aP₀) atr₀, which is the closest to the aperture, and a portion of a cone with a vertex atP₀, which is labeledA₄ in the right diagram. If the wavefront is positioned such that the wavefront is very close to the edges of the aperture, then the contribution fromA₄ can be neglected (assumed here). On this newA₁, the inward (toward the volume enclosed by the closed integral surface, so toward the right side in the diagram) normal${\displaystyle n}$ toA₁ is along the radial direction fromP₀, i.e., the direction perpendicular to the wavefront. As a result, the angle${\displaystyle (n,r)=0}$ and the angle${\displaystyle (n,s)}$ is related with the angle${\displaystyle \chi }$ (the angle as defined inHuygens–Fresnel principle) as$${\displaystyle (n,s)=\pi -\chi .}$$

The complex amplitude of the wavefront atr₀ is given by$${\displaystyle U(r_0)=\frac{a{e}^{ik{r}_0}}{r_0}.}$$

So, the diffraction formula becomes$${\displaystyle U(P)=-\frac{i}{2\lambda }\frac{a{e}^{ik{r}_0}}{r_0}{\int }_S\frac{e^{iks}}{s}(1+\cos \chi )dS,}$$where the integral is done over the part of the wavefront atr₀ which is the closest to the aperture in the diagram. This integral leads to theHuygens–Fresnel principle (with the obliquity factor$\frac{1+\cos \chi }{2}$).

In the derivation of this integral, instead of the geometry depicted in the right diagram, double spheres centered atP₀ with the inner sphere radiusr₀ and an infinite outer sphere radius can be used.^[7] In this geometry, the observation pointP is located in the volume enclosed by the two spheres so the Fresnel-Kirchhoff diffraction formula is applied on the two spheres. (The surface normal on these integral surfaces are, say again, toward the enclosed volume in the diffraction formula above.) In the formula application, the integral on the outer sphere is zero by a similar reason of the integral on the hemisphere as zero above.

Assume that the aperture is illuminated by an extended source wave.^[8] The complex amplitude at the aperture is given byU₀(r).

It is assumed, as before, that the values of${\displaystyle U}$ and${\displaystyle \frac{\mathrm{\partial }U}{\mathrm{\partial }n}}$ in the areaA₁ are the same as when the screen is not present, that the values of${\displaystyle U}$ and${\displaystyle \frac{\mathrm{\partial }U}{\mathrm{\partial }n}}$ inA₂ are zero (Kirchhoff's boundary conditions) and that the contribution fromA₃ to the integral are also zero. It is also assumed that 1/s is negligible compared withk. We then have$${\displaystyle U(P)=\frac{1}{4\pi }{\int }_{A_1}\frac{e^{iks}}{s}\left[ik{U}_0(r)\cos (n,s)-\frac{\mathrm{\partial }{U}_0(r)}{\mathrm{\partial }n}\right]dS.}$$

This is the most general form of the Kirchhoff diffraction formula. To solve this equation for an extended source, an additional integration would be required to sum the contributions made by the individual points in the source. If, however, we assume that the light from the source at each point in the aperture has a well-defined direction, which is the case if the distance between the source and the aperture is significantly greater than the wavelength, then we can write$${\displaystyle U_0(r)\approx a(r)e^{ikr},}$$wherea(r) is the magnitude of the disturbance at the pointr in the aperture. We then have$${\displaystyle \frac{\mathrm{\partial }{U}_0(r)}{\mathrm{\partial }n}=ika(r)e^{ikr}\cos (n,r)}$$and thus$${\displaystyle U(P)=-\frac{i}{2\lambda }{\int }_{S}a(r)\frac{e^{ik(s+r)}}{s}[\cos \chi +\cos (n,r)]dS.}$$

In spite of the various approximations that were made in arriving at the formula, it is adequate to describe the majority of problems in instrumental optics. This is mainly because the wavelength of light is much smaller than the dimensions of any obstacles encountered. Analytical solutions are not possible for most configurations, but theFresnel diffraction equation andFraunhofer diffraction equation, which are approximations of Kirchhoff's formula for thenear field andfar field, can be applied to a very wide range of optical systems.

One of the important assumptions made in arriving at the Kirchhoff diffraction formula is thatr ands are significantly greater than λ. Another approximation can be made, which significantly simplifies the equation further: this is that the distancesP₀Q andQP are much greater than the dimensions of the aperture. This allows one to make two further approximations:

• cos(n, r) − cos(n, s) is replaced with 2cos β, where β is the angle betweenP₀P and the normal to the aperture. The factor 1/rs is replaced with 1/r's', wherer' ands' are the distances fromP₀ andP to the origin, which is located in the aperture. The complex amplitude then becomes:$${\displaystyle U(P)=-\frac{ia\cos \beta }{\lambda r^{\prime }{s}^{\prime }}{\int }_S{e}^{ik(r+s)}ds.}$$
• Assume that the aperture lies in thexy plane, and the coordinates ofP₀,P andQ (a general point in the aperture) are (x₀,y₀,z₀), (x,y,z) and (x',y', 0) respectively. We then have:

  ${\displaystyle r^2=(x_0-x^{\prime }{)}^2+(y_0-y^{\prime }{)}^2+z_0^2,}$

  ${\displaystyle s^2=(x-x^{\prime }{)}^2+(y-y^{\prime }{)}^2+z^2,}$

  ${\displaystyle r^{\prime 2}=x_0^2+y_0^2+z_0^2,}$

  ${\displaystyle s^{\prime 2}=x^2+y^2+z^2.}$

We can expressr ands as follows:$${\displaystyle r=r^{\prime }{\left[1-\frac{2(x_0{x}^{\prime }+y_0{y}^{\prime })}{r^{\prime 2}}+\frac{x^{\prime 2}+y^{\prime 2}}{r^{\prime 2}}\right]}^{1/2},}$$$${\displaystyle s=s^{\prime }{\left[1-\frac{2(x{x}^{\prime }+y{y}^{\prime })}{s^{\prime 2}}+\frac{x^{\prime 2}+y^{\prime 2}}{s^{\prime 2}}\right]}^{1/2}.}$$

These can be expanded aspower series:$${\displaystyle r=r^{\prime }\left[1-\frac{1}{2r^{\prime 2}}\left[2(x_0{x}^{\prime }+y_0{y}^{\prime })-(x^{\prime 2}+y^{\prime 2})\right]+\frac{1}{8r^{\prime 4}}{\left[2(x_0{x}^{\prime }+y_0{y}^{\prime })-(x^{\prime 2}+y^{\prime 2})\right]}^2+\cdots \right],}$$$${\displaystyle s=s^{\prime }\left[1-\frac{1}{2s^{\prime 2}}\left[2(x{x}^{\prime }+y{y}^{\prime })-(x^{\prime 2}+y^{\prime 2})\right]+\frac{1}{8s^{\prime 4}}{\left[2(x{x}^{\prime }+y{y}^{\prime })-(x^{\prime 2}+y^{\prime 2})\right]}^2+\cdots \right].}$$

The complex amplitude atP can now be expressed as$${\displaystyle U(P)=-\frac{i\cos \beta }{\lambda }\frac{a{e}^{ik(r^{\prime }+s^{\prime })}}{r^{\prime }{s}^{\prime }}{\int }_S{e}^{ikf(x^{\prime },y^{\prime })}d{x}^{\prime }d{y}^{\prime },}$$wheref(x',y') includes all the terms in the expressions above fors andr apart from the first term in each expression and can be written in the form$${\displaystyle f(x^{\prime },y^{\prime })=c_1{x}^{\prime }+c_2{y}^{\prime }+c_3{x}^{\prime 2}+c_4{y}^{\prime 2}+c_5{x}^{\prime }{y}^{\prime }\cdots ,}$$where thec_i are constants.

If all the terms inf(x',y') can be neglected except for the terms inx' andy', we have theFraunhofer diffraction equation. If the direction cosines ofP₀Q andPQ are

TheFraunhofer diffraction equation is then$${\displaystyle U(P)=C{\int }_S{e}^{ik[(l_0-l)x^{\prime }+(m_0-m)y^{\prime }]}d{x}^{\prime }d{y}^{\prime },}$$whereC is a constant. This can also be written in the form$${\displaystyle U(P)=C{\int }_S{e}^{i({\mathbf{k}}_0-\mathbf{k})\cdot {\mathbf{r}}^{\prime }}d{r}^{\prime },}$$wherek₀ andk are thewave vectors of the waves traveling fromP₀ to the aperture and from the aperture toP respectively, andr' is a point in the aperture.

If the point source is replaced by an extended source whose complex amplitude at the aperture is given byU₀(r'), then theFraunhofer diffraction equation is:$${\displaystyle U(P)\propto {\int }_S{a}_0({\mathbf{r}}^{\prime })e^{i({\mathbf{k}}_0-\mathbf{k})\cdot {\mathbf{r}}^{\prime }}d{r}^{\prime },}$$wherea₀(r') is, as before, the magnitude of the disturbance at the aperture.

In addition to the approximations made in deriving the Kirchhoff equation, it is assumed that

• r ands are significantly greater than the size of the aperture,
• second- and higher-order terms in the expressionf(x',y') can be neglected.

When the quadratic terms cannot be neglected but all higher order terms can, the equation becomes theFresnel diffraction equation. The approximations for the Kirchhoff equation are used, and additional assumptions are:

• r ands are significantly greater than the size of the aperture,
• third- and higher-order terms in the expressionf(x',y') can be neglected.

• Baker, B.B.; Copson, E.T. (1939, 1950).The Mathematical Theory of Huygens' Principle. Oxford.
• Woan, Graham (2000).The Cambridge Handbook of Physics Formulas. Cambridge University Press.ISBN 9780521575072.
• J. Goodman (2005).Introduction to Fourier Optics (3rd ed.). Roberts & Co Publishers.ISBN 978-0-9747077-2-3.
• Griffiths, David J. (2012).Introduction to Electrodynamics. Pearson Education, Limited.ISBN 978-0-321-85656-2.
• Band, Yehuda B. (2006).Light and Matter: Electromagnetism, Optics, Spectroscopy and Lasers.John Wiley & Sons.ISBN 978-0-471-89931-0.
• Kenyon, Ian (2008).The Light Fantastic: A Modern Introduction to Classical and Quantum Optics.Oxford University Press.ISBN 978-0-19-856646-5.
• Lerner, Rita G.; George L., Trigg (1991).Encyclopedia of physics. VCH.ISBN 978-0-89573-752-6.
• Sybil P., Parker (1993).MacGraw-Hill Encyclopedia of Physics. McGraw-Hill Ryerson, Limited.ISBN 978-0-07-051400-3.

• Waves
• Physical optics
• Diffraction
• Gustav Kirchhoff

• CS1 German-language sources (de)
• Webarchive template wayback links
• Articles with short description
• Short description matches Wikidata