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Kirchhoff's diffraction formula

Kirchhoff’s diffraction formula can be used to model the propagation of light in a wide range of configurations, either analytically or using numerical modelling. It uses the Kirchhoff integral theorem which applies Green's theorem to derive a solution to the general time independent homogenuous wave equation. Kirchhoff calculates the wave disturbance when a spherical wave passes through an opening in an opaque screen.

Kirchhoff's theorem

Kirchhoff's theorem (sometimes referred to as the Fresnel–Kirchhoff integral theorem) provides an approximate solution to the homogeneous wave equation at a specific point, P, in terms of the conditions of the solution and its first order derivative at all points on an arbitrary surface which encloses the point. The form of the equation for a monochromatic source is:

$U(P)={\frac {1}{4\pi }}\int _{S}\left[U{\frac {\partial }{\partial n}}\left({\frac {e^{iks}}{s}}\right)-{\frac {e^{iks}}{s}}{\frac {\partial U}{\partial n}}\right]dS$

where k is the wavenumber and s is the distance from P to the surface.

Geometrical arrangement used in deriving Kirchhhoff's diffraction formula

Consider a monochromatic point source at P₀ which illuminates an aperture in a screen. The disturbance at a point P can be found by applying the integral theorem to the closed surface which is formed by the intersection of a circle(sphere?) of radius R with the screen. The integration is performed over the areas A₁, A₂ and A₃ giving

$U(P)={\frac {1}{4\pi }}\left[\int _{A_{1}}+\int _{A_{2}}+\int _{A_{3}}\left[U{\frac {\partial }{\partial n}}\left({\frac {e^{iks}}{s}}\right)-{\frac {e^{iks}}{s}}{\frac {\partial U}{\partial n}}\right]\right]dS$

To solve the equation, it is assumed that the values of U and ∂U/∂n in the area A₁ are the same as when the screen is not present, giving:

$U_{A_{1}}={\frac {ae^{ikr}}{r}}$

${\frac {\partial U_{A_{1}}}{\partial n}}={\frac {ae^{ikr}}{r}}\left[ik-{\frac {1}{r}}\right]\cos {(n,r)}$

?????? where a represents the magnitude of the disturbance at P₀, r is the distance between P₀, shown in the diagram for a particular point in the aperture, Q, and (n,r) is the angle between r and the normal to the aperture.

Kirchoff assumes that the values of U and ∂U/∂n in A₂ are zero. This implies that U and ∂U/∂n are discontinuous at the edge of the aperture. This is not the case, and this is one of the approximation used in deriving the equation.

???The contribution from A₃ to the integral is also assumed to be zero. This can be justified by making the assumption that the source starts to radiate at a particular time, and then by making 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; but that assumption is not necessary and a more formal argument avoiding its use has been found.

Part 1:Finally, the terms 1/r and 1/s are assumed to be negligible compared with k since r and s are generally much greater than λ, which is equal to 2π/k. Part 2 Thus the integral above which represents the complex amplitude at P becomes:

$U(P)=-{\frac {ia}{2\lambda }}\int _{S}{{\frac {e^{ik(r+s)}}{rs}}[\cos(n,r)-\cos(n,s)]}dS$

where (n,s) is the angle between the normal to the aperture and s. This is the Kirchhoff or Fresnel–Kirchhoff diffraction formula.

??? Geometric arrangment used to express Kirchhoff's formula in a form similar to Huygens–Fresnel

The formula can be expressed in a form similar to the Huygens–Fresnel principle by using a different closed surface over which the integration is performed. The area A₁ above is replaced by a wavefront from P₀ which almost fills the aperture, and a portion of a cone with a vertex at P₀ which is labelled A₄ in the diagram. If the radius of curvature of the wave is large enough, the contribution from A₄ can be neglected. We also have where χ is as defined in [[Huygens–Fresnel principle] and cos(n,r) = 1. The complex amplitude of the wavefront at r₀ is given by:

$U(r_{0})={\frac {ae^{ikr_{0}}}{r_{0}}}$

The diffraction formula becomes;

$U(P)=-{\frac {i}{2\lambda }}{\frac {ae^{ikr_{0}}}{r_{0}}}\int _{S}{\frac {e^{iks}}{s}}(1+\cos \chi )\,dS$

This is the Kirchhoff's diffraction formula which contains parameters which had to be arbitrarily assigned in the derivation of the Huygens–Fresnel equation.

In spite of the various approximations which 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. The Fresnel diffraction equation and Fraunhofer diffraction equation are approximations of Kirchhoff’s formula for the near field and far field and can be applied to a very wide range of optical configurations.

Fraunhofer and Fresnel approximations

(Under development)

One of the important assumptions made in arriving at the Kirchhoff diffraction formula is that r and s are signficantly greater than λ. A further approximation can be made which signficantly simplifies the equation further: this is that the distance of P₀ and P are much greater than the dimensions of the aperture. This allows us to make tow further approximations:

cos(n,r)-cos(n,s) is replaced with 2cos β where β is the angle between P₀P

The factor 1/rs is replaced with 1/r₀s₀ where r₀, s₀ are the distances from P₀ and P to the origin, which is located in the aperture. The complex amplitude then becomes:

$U(P)=-{\frac {ia\cos \beta }{\lambda r's'}}\int _{S}e^{ik(r+s)}dS$

We assume that the aperture lies in the yz plane, and the co-ordinates of P₀, P and Q (a general point in the aperture) are (x₀,y₀,x₀), (x,y,z) and (0,y' ,z' ) respectively. We then have:

$~r^{2}={(x_{0}-x')^{2}+(y_{0}-y')^{2}+z_{0}^{2}}$

$~s^{2}={(x-x')^{2}+(y-y')^{2}+z^{2}}$

$~r'^{2}=x_{0}^{2}+y_{0}^{2}+z_{0}^{2}$

$~s'^{2}=x^{2}+y^{2}+z^{2}$

We can express r and s as follows:

$r=r'\left[{1-{\frac {2(x_{0}x'+y_{0}y')}{r'}}+{\frac {x'^{2}+y'^{2}}{r'}}}\right]^{1/2}$

$s=s'\left[{1-{\frac {2(xx'+yy')}{s'}}+{\frac {x'^{2}+y'^{2}}{s'}}}\right]^{1/2}$

We can expand these as power series:

$r=r'\left[{1-{\frac {1}{2r'}}[2(x_{0}x'+y_{0}y')+(x'^{2}+y'^{2})]+{\frac {1}{2r'}}[2(x_{0}x'+y_{0}y')+(x'^{2}+y'^{2})]^{2}+........}\right]$

$s=s'\left[{1-{\frac {1}{2s'}}[2(xx'+yy')+(x'^{2}+y'^{2})]+{\frac {1}{2s'}}[2(xx'+yy')+(x'^{2}+y'^{2})]^{2}+........}\right]$

The complex amplitude at P can now be expressed as:

$U(P)=-{\frac {i\cos \beta }{\lambda }}{\frac {ae^{ik(r'+s')}}{r's'}}\int _{S}e^{ikf(x',y')}dx'dy'$

where f(x',y') inlcudes all the terms in the expressions above for s and r apart from the first term in each expression and can be written in the form:

$~f(x'y')=c_{1}x'+c_{2}y'+c_{3}x'^{2}+c_{4}y'^{2}+c_{5}x'y'.......$

where the c are constants.

If all the terms can be neglected except for the terms in x' and y' , we have the Fraunhofer diffraction equation. If the direction cosines of P₀Q and PQ are

$l_{0}=-{\frac {x_{0}}{r'}}$, $m_{0}=-{\frac {y_{0}}{r'}}$

$l={\frac {x}{s'}}$, $m={\frac {y}{s'}}$

The Fraunhofer diffraction equation is then

$U(P)=C\int _{S}e^{ik[(l_{0}-l)x'+(m-m_{0})y']}dx'dy'$

where C is a constant.

When the quadratic terms cannot be neglected but all higher order terms can, the equation becomes the Fresnel diffraction equation.

References

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