Kirchhoff's diffraction formula (also called Fresnel–Kirchhoff diffraction formula) approximates light intensity and phase in optical diffraction: light fields in the boundary regions of shadows. The approximation can be used to model light propagation in a wide range of configurations, either analytically or using numerical modelling. It gives an expression for the wave disturbance when a monochromatic spherical wave is the incoming wave of a situation under consideration. This formula is derived by applying the Kirchhoff integral theorem, which uses the Green's second identity to derive the solution to the homogeneous scalar wave equation, to a spherical wave with some approximations.
The Huygens–Fresnel principle can be derived by the Fresnel–Kirchhoff diffraction formula.
Derivation of Kirchhoff's diffraction formula
Kirchhoff's integral theorem, sometimes referred to as the Fresnel–Kirchhoff integral theorem, uses Green's second identity to derive the solution of the homogeneous monochromatic scalar wave equation at an arbitrary spatial position P in terms of the solution of the wave equation and its first order derivative at all points on an arbitrary closed surface as the boundary of some volume including P.
The solution provided by the integral theorem for a monochromatic source is
where is the spatial part of the solution of the homogeneous scalar wave equation (i.e., as the homogeneous scalar wave equation solution), k is the wavenumber, and s is the distance from P to an (infinitesimally small) integral surface element, and is a normal derivative, that is a derivative of the function along the unit normal vector of an integral surface element . The surface normal unit vector is toward the inside of the enclosed volume in this integral.
In deriving the Kirchoff's diffraction formula, following assumptions are made. The distance between a point source P0 of a spherical wave and an integral area, and the distance between the integral area and an observation point P, are much greater than the wave wavelength . A wave on the aperture is same to the wave that would be present if there was no wave obstacle.This assumption may be expressed differently: the dimension of opening (aperture) is much greater than the wave wavelength . and are zero at the obstacle. The 2nd and 3rd conditions are called Kirchhoff's boundary conditions, and they together lead to a physically incorrectly condition that the calculated wave is discontinuous at the aperture boundaries.
Point source Consider a monochromatic spherical wave point source at P0, which illuminates an aperture A1 in a screen as shown in the right figure. The intensity of the wave emitted by the 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 is given by
where represents the magnitude of the disturbance at the point source.
The disturbance at a spatial position P can be found by applying the Kirchhoff's integral theorem to the closed surface formed by the screen, consisted of A1 and A2, and a unblocked part of a sphere of radius R centered at C, A3. The integration is performed over the areas A1, A2 and A3, giving
To solve the equation, it is assumed that the values of and in the aperture area A1 are the same as when the screen is not present (1st assumption of the Kirchhoff's diffraction formula, one of two Kirchhoff's boundary conditions), so at the position Q on the aperture,
The del operator is in terms of independent variables of the functions and . Here .where is the length of the straight line P0Q, and is the angle between a straightly extended version of P0Q (the extension is directed from P0 to Q) and the (inward) unit normal vector to the aperture. Note that so is a positive real number on A1.
At Q, we also have
where is the length of the straight line PQ, and is the angle between a straightly extended version of PQ (the extension is directed from P to Q) and the (inward) normal to the aperture. Note that so is a negative real number on A1.
Two more following assumptions are made.
(2nd assumption of the Kirchhoff's diffraction formula) In the above normal derivatives, the terms and in both square brackets are assumed to be negligible compared with the wavenumber , means and are much greater than the wavelength . Thus, (3rd assumption of the Kirchhoff's diffraction formula, one of the two Kirchhoff's boundary conditions) Kirchhoff assumes that the values of and on the opaque areas marked by A2 are zero. This implies that and are discontinuous at the edge of the aperture A1. Of course this is generally not the case.J.Z. Buchwald & C.-P. Yeang, "Kirchhoff's theory for optical diffraction, its predecessor and subsequent development: the resilience of an inconsistent theory" , Archive for History of Exact Sciences, vol.70, no.5 (Sep.2016), pp.463–511; .J. Saatsi & P. Vickers, "Miraculous success? Inconsistency and untruth in Kirchhoff’s diffraction theory", British J. for the Philosophy of Science, vol.62, no.1 (March 2011), pp. 29–46; jstor.org/stable/41241806. (Pre-publication version, with different pagination: dro.dur.ac.uk/10523/1/10523.pdf.)
The contribution from A3 to the integral is expected to be zero, and it can be justified by one of the following reasons. (The 2nd one is preferred as the 1st one has a physical flaw.) Make the assumption that the source starts to radiate at a particular time, and make R large enough, so that when the disturbance at P is being considered, no contributions from A3 will have arrived there. Such a wave is no longer monochromatic, since a monochromatic wave must exist at all times, while the monochromaticity is assumed in the Kirchoff's diffraction formula. This physically flawed assumption is not necessary, and a more formal argument avoiding it has been derived.M. Born, Optik: ein Lehrbuch der elektromagnetischen Lichttheorie. Berlin, Springer, 1933, reprinted 1965, p. 149. A wave emanated from the aperture A1 is expected to evolve toward a spherical wave as it propagates.Water wave examples of this can be found in many pictures showing that a water wave passes through a relatively narrow opening and then the wave evolves to a spherical wave as it propagates away from the opening. So, if R is large enough, then the wave on A3 is with being the distance from the center of the aperture A1, thus the integral on A3 becomeswhere is the distance from the center of the aperture A1 to an integral surface element.
As a result, finally, the integral above, which represents the complex amplitude at P, becomes
This is the Kirchhoff or Fresnel–Kirchhoff diffraction formula.
This formula can also be derived by using a different geometry in which non-physical (no so wave blocking) arbitrary closed surface, that is apart from the point wave source P0 and surrounds the observation point P, is the integral surface to which the Kirchhoff's integral theorem is applied. In this geometry, among three assumptions made above, only the 2nd assumption - (The distance between P0 and an element of the integral surface) and (The distance between the integral surface element and P) are much greater than the wavelength - is required. The result is the same integral form of the Fresnel–Kirchhoff diffraction formula above, except that the integral surface is the chosen closed surface.
Equivalence to Huygens–Fresnel principle
The Huygens–Fresnel principle can be derived by integrating over a different closed surface (the boundary of some volume having an observation point P). In the right diagram, the aperture area A1 in the diagram of the above section is replaced by (1) a part of a wavefront (emitted from a P0) at r0, which is the closest to the aperture, and (2) a portion of a cone with a vertex at P0, labelled A4, that is straight lines perpendicular to the wavefront and between the wavefront and the edges of the aperture. If the wavefront is positioned such that the wavefront is very close to the edges of the aperture, then the contribution from A4 can be neglected (an additional assumption on the top of three assumptions used to derive the Kirchhoff's diffraction formula above). On this new A1, the inward (toward the volume enclosed by the closed integral surface, so toward the right side in the diagram) normal to A1 is along the radial direction from P0, i.e., the direction perpendicular to the wavefront. As a result, the angle and the angle is related with the angle (the angle as defined in Huygens–Fresnel principle) as
The complex amplitude of the wavefront at r0 is given by
So, the diffraction formula in the above section becomes
where the integral is done over the part of the wavefront at r0 which is the closest to the aperture in the diagram. This integral leads to the Huygens–Fresnel principle (with the obliquity factor ).
In the derivation of this integral, instead of the geometry depicted in the diagram of this section, two closed surfaces (no physical surfaces so they do not block waves), one is a sphere of radius r0 centered at the point wave source P0, and the other is an arbitrary closed surface that contains this inner sphere and is infinitely far from it (e.g., another sphere of infinite radius), can be used. In this geometry, the observation point P is located in the volume enclosed by these closed surfaces, 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 formula application to this geometry, the additional assumption of neglecting the integral on A4, used in the above diagram of this section, is not required. The integral on the outer surface is zero as long as it is infinitely far from P, so as a result, the Huygens–Fresnel principle in the same integral from above is derived, except that the integral surface is the inner sphere.
Extended source Assume that an aperture is illuminated by waves from an extended monochromatic wave source.M. V. Klein & T. E. Furtak, 1986, Optics; 2nd ed. John Wiley & Sons, New York . The complex amplitude at the aperture is given by .In a point wave source case above, it is .
It is assumed, as before, that the values of and in the area A1 are the same as when a screen having the aperture is not present, that the values of and in A2 are zero (Kirchhoff's boundary conditions), and that 1/s is negligible compared with a wavenumber k (s is much greater than the wavelength )This is a part of the 2nd assumption used to derive the Kirchhoff's diffraction formula for a spherical wave point source.. By the same logic as before, the integral over A3 is zero. By applying the Kirchhoff's integral theorem to A1, we then have
This is the most general form of the Kirchhoff diffraction formula.
To solve this equation for an extended wave source, an additional integration would be required to sum contributions from individual points in the source. If, however, we assume that the distance between the source and the aperture is significantly greater than the wavelengthThis is the remaining part of the 2nd assumption used to derive the Kirchhoff's diffraction formula for a spherical wave point source., then we may approximately write a wave in the aperture like a spherical wave (It may be an additional assumption on the top of three assumptions used to derive the Kirchhoff's diffraction formula for a point wave source above.)
where is the magnitude of the disturbance at the point in the aperture. We then have