Algorithms for Fresnel Diffraction at Rectangular and Circular Apertures

Abstract

This paper summarizes the theory of Fresnel diffraction by plane rectangular and circular apertures with a view toward numerical computations. Approximations found in the earlier literature, and now obsolete, have been eliminated and replaced by algorithms suitable for use on a personal computer.

1. Introduction

The basic theory of Fresnel diffraction at plane apertures was developed long ago [1,2] and is summarized in textbooks [3–6]. For apertures bounded by straight lines (rectangle, slit, half plane), the standard textbook solution in terms of complex Fresnel integrals1 is based on special but poorly documented transformations of coordinates. It is shown in this paper that such transformations cannot be performed accurately for apertures irradiated by arbitrarily located point sources. In the past, the numerical evaluation of the complex Fresnel integrals themselves has also been a problem, and thus previous discussions were confined to simplified special cases. Computational details were omitted and semi-quantitative methods (Cornu spiral) were used to describe the nature of diffraction at rectangular apertures.

In the case of circular apertures, the rigorous solution involves Lommel functions of two variables, which are defined as series expansions in Bessel functions and previously had to be evaluated by tedious manual calculations or approximations. For the most part, these approaches have been rendered obsolete by modern computer software. However, approximative methods are still useful for work on personal computers which involves large values of the configuration parameter u defined by Eq. (17a) of this paper. It is shown here that a previously used approximation by Focke [7] is inadequate for this purpose on account of its poor accuracy, but that an older approximation by Schwarzschild [8] gives excellent results.

As algorithms for the computation of Fresnel diffraction patterns on a personal computer have not been published, a compilation of such algorithms is presented in this paper. The underlying theory is stated for off-axis source points, so that the results can be applied to extended sources. For both types of aperture, the closed solutions obtained are paraxial approximations.

2. The Fresnel Diffraction Integral

The scalar wave function U(P) associated with diffraction at a plane aperture is customarily expressed by one of the Rayleigh-Sommerfeld integrals2,

$$U_{\text{RS}}^{\mathrm{p}}(P)=-\frac{A_{\text{sph}}}{2\mathrm{\pi }}{\displaystyle \int \mathrm{d}Q\frac{{\mathrm{e}}^{\mathrm{i}k(P_{0}Q+QP)}}{P_{0}Q\cdot QP}}\left(\mathrm{i}k-\frac{1}{P_{0}Q}\right)\frac{\partial P_{0}Q}{\partial \mathit{n}},$$ (1a)

$$U_{\text{RS}}^{\mathrm{s}}(P)=-\frac{A_{\text{sph}}}{2\mathrm{\pi }}{\displaystyle \int \mathrm{d}Q\frac{{\mathrm{e}}^{\mathrm{i}k(P_{0}Q+QP)}}{P_{0}Q\cdot QP}}\left(\mathrm{i}k-\frac{1}{QP}\right)\frac{\partial QP}{\partial \mathit{n}},$$ (1b)

or, alternatively, by the Kirchhoff integral,

$$U_{\mathrm{K}}(P)=\frac{1}{2}(U_{\text{RS}}^{\mathrm{p}}+U_{\text{RS}}^{\mathrm{s}}).$$ (1c)

Here, as indicated in Figs. 1, 2, and 4, P₀ is the location of a point source emitting a monochromatic spherical wave of amplitude A_sph, circular wave number k = 2π/λ Q is a point in the aperture, dQ is the surface element at Q, n is the aperture normal pointing away from the source, and P is the point of observation.

Fig. 1.

Notation for rectangular apertures.

Fig. 2.

Notation for slits.

Fig. 4.

Notation for circular apertures.

In the Fresnel approximation the points P₀ and P are located at finite distances which are large compared to the wavelength of light and the dimensions of the aperture. Therefore, it is assumed that

$$\frac{1}{P_{0}Q}<<k,\frac{1}{QP}<<k,$$ (2a)

and that the distances P₀Q and QP and their normal derivatives do not vary appreciably inside the aperture. Thus they can be replaced, except in the rapidly oscillating exponential function in the integrand, by their values at an arbitrarily chosen reference point O inside the aperture. Under these conditions, the first Rayleigh-Sommerfeld integral Eq. (1a) may be written in the form

$$U_{\text{RS}}^{\mathrm{p}}(P)\approx -\frac{\mathrm{i}k{A}_{\text{sph}}}{2\mathrm{\pi }}\frac{\partial P_{0}O}{\partial \mathit{n}}\frac{{\mathrm{e}}^{\mathrm{i}k(P_{0}O+OP)}}{P_{0}O\cdot OP}{\displaystyle \int \mathrm{d}Q{\mathrm{e}}^{\mathrm{i}k\mathrm{\Delta }(Q)}},$$ (2b)

where

$$\mathrm{\Delta }(Q)=(P_{0}Q+QP)-(P_{0}O+OP),$$ (2c)

is a small quantity that can be expressed in approximate form. It is well known that, with

$$O=(0,0,0),P_0=(x_0,y_0,z_0),Q=(\xi ,\eta ,0),P=(x,y,z)$$ (2d)

expressed in cartesian coordinates, the required approximation for Δ(Q) is

$$\begin{array}{l}\mathrm{\Delta }(Q)=-[(l-l_0)\xi +(m-m_0)\eta ]+\frac{1}{2r_0}[({\xi }^2+{\eta }^2)-\\ {\left(l_0\xi +m_0\eta \right)}^2]+\frac{1}{2r}[({\xi }^2+{\eta }^2)-{\left(l\xi +m\eta \right)}^2]+ϵ(\xi ,\eta ),\end{array}$$ (2e)

where ϵ (ξ,η) is the residual error when terms of third and higher order in ξ and η are neglected. Here,

$$l_0=-\frac{x_0}{r_0},m_0=-\frac{y_0}{r_0},l=\frac{x}{r},m=\frac{y}{r}$$ (2f)

are the first and second direction cosines of the vectors P₀O and OP, and

$$r_0=\sqrt{x_0^2+y_0^2+z_0^2},r=\sqrt{x^2+y^2+z^2}$$ (2g)

are the distances P₀O and OP. The corresponding value of the normal derivative3 in Eq. (2b) is

$$\frac{\partial P_{0}O}{\partial \mathit{n}}=-\frac{\partial r_0}{\partial z_0}=-\frac{z_0}{r_0}=-\cos {\theta }_0.$$ (2h)

We now have

$$U_{\text{RS}}^{\mathrm{p}}(P)\approx \frac{\mathrm{i}k{A}_{\text{sph}}\cos {\theta }_0}{2\mathrm{\pi }{r}_{0}r}{\mathrm{e}}^{\mathrm{i}k(r_0+r)}{\displaystyle \int \mathrm{d}Q{\mathrm{e}}^{\mathrm{i}k\mathrm{\Delta }(Q)}}.$$ (2i)

The corresponding forms of $U_{\text{RS}}^{\mathrm{s}}(P)$ and U_K(P) are essentially the same, except that –cosθ₀ is replaced by cosθ and ½ (−cosθ₀ + cosθ), respectively, where θ is the colatitude of the point of observation P. Because the diffracted light is confined to a narrow angular range about the central direction P₀Q unless the aperture dimensions are extraordinarily small, these differences may be judged insignificant. As the Rayleigh-Sommerfeld solutions pertain to the respective cases of p- and s-polarization of the incident light, this implies that Fresnel diffraction is independent of polarization. The Kirchhoff solution has no definable meaning as far as polarization is concerned, but turns out to be equivalent to the Rayleigh-Sommerfeld solutions in the Fresnel approximation. A further solution, the Maggi-Rubinowicz transformation of Kirchhoff’s integral [3,4], is not suitable for computations of Fresnel diffraction patterns because it is singular at the boundary of the geometrical shadow.

In this paper, Eq. (2i) will be regarded as the basic form of the Fresnel diffraction integral and will be written as

$$U_{\mathrm{F}}(P)=-U_0(P)\cos {\theta }_0{\alpha }_{\mathrm{F}}(P),$$ (3a)

where

$$U_0(P)=A_{\text{sph}}\frac{{\mathrm{e}}^{\mathrm{i}k(r_0+r)}}{r_0+r}=\sqrt{E_0(P)}{\mathrm{e}}^{\mathrm{i}k(r_0+r)}$$ (3b)

is the geometrical field at the point of observation P according to Huygens’ principle,

$${\alpha }_{\mathrm{F}}(P)=-\frac{\mathrm{i}k(r_0+r)}{2\mathrm{\pi }{r}_{0}r}{\displaystyle \int \mathrm{d}Q{\mathrm{e}}^{\mathrm{i}k\mathrm{\Delta }(Q)}}$$ (3c)

is the modification of the geometrical field by diffraction, E₀(P) is the normally incident geometrical irradiance at P, and –cosθ₀ is the inclination factor according to Lambert’s law.

The third- and fourth-order terms neglected in Eq. (2e) are

$$\begin{array}{l}ϵ(\xi ,\eta )=-\frac{1}{2r_0^2}\{(l_0\xi +m_0\eta )[({\xi }^2+{\eta }^2)-{\left(l_0\xi +m_0\eta \right)}^2]\}\\ +\frac{1}{2r^2}\{(l\xi +m\eta )[({\xi }^2+{\eta }^2)-{\left(l\xi +m\eta \right)}^2]\}\\ -\frac{1}{8r_0^3}\{{\left({\xi }^2+{\eta }^2\right)}^2-{(l_0\xi +m_0\eta )}^2[6({\xi }^2+{\eta }^2)\\ -5{\left(l_0\xi +m_0\eta \right)}^2]\}\\ -\frac{1}{8r^3}\{{\left({\xi }^2+{\eta }^2\right)}^2-{\left(l\xi +m\eta \right)}^2[6({\xi }^2+{\eta }^2)\\ -5{\left(l\xi +m\eta \right)}^2]\}.\end{array}$$ (4a)

This shows that the error introduced by neglecting this term depends in a complicated manner on the geometrical parameters involved. Accordingly, it is difficult to assess its magnitude without considering specific cases. However, a few general comments are in order. Under ordinary circumstances, the direction cosines l₀, l, … are small compared with unity, so that (l₀ξ + m₀η)² and (lξ + mη)² are much smaller than (ξ² + η²), and then one finds4

$$ϵ(\xi ,\eta )\approx -\frac{r_0^3+r^3}{8r_0^3{r}^3}{\left({\xi }^2+{\eta }^2\right)}^2.$$ (4b)

Hence, the magnitude of ϵ (ξ,η) relative to the quadratic term of Eq. (2e) may be estimated as

$$\frac{2r_{0}r/ϵ(\xi ,\eta )|}{(r_0+r)({\xi }^2+{\eta }^2)}\approx \frac{(r_0^3+r^3)({\xi }^2+{\eta }^2)}{4r_0^3{r}^2(r_0+r)}<\frac{q_{\max }^2}{{\langle r\rangle }^2},$$ (4c)

where q_max is the maximum value of $\sqrt{{\xi }^2+{\eta }^2}$(e.g., the radius of a circular aperture) and 〈r〉 is an average of r₀ and r. Accordingly, the relative error in Δ(Q) is inversely proportional to the square of the relative distance 〈r〉/q_max. At a distance of ten aperture dimensions, it is on the order of 1 %.

3. Rectangular Aperture

3.1 General Theory (Fig. 1)

When applying the above equations to a rectangular aperture of width 2w and height 2h, it is customary to transform the global cartesian coordinates (x, y, z) assumed in Sec. 2 into local coordinates (x′, y′, z′) which depend on the locations of the points P₀ and P and are chosen so that Eq. (3c) is separated into a product of independent Fresnel integrals in ξ and η. That is,

$${\alpha }_{\mathrm{F}}(P)\propto {\displaystyle \int \mathrm{d}\xi {\mathrm{e}}^{\mathrm{i}a{\xi }^2}}{\displaystyle \int \mathrm{d}\eta {\mathrm{e}}^{\mathrm{i}b{\eta }^2}}.$$ (5)

The first step in this transformation is to place the origin of the local coordinates at the point M where the straight line P₀P intersects the aperture plane:

$$M=(x_M,y_M,0),x_M=\frac{x_{0}z-x{z}_0}{z-z_0},y_M=\frac{y_{0}z-y{z}_0}{z-z_0}.$$ (6a)

This gives

$$\begin{array}{c}{l_0}^{\prime }=\frac{x_0-x_M}{{r_0}^{\prime }}=l^{\prime }=\frac{x-x_M}{r^{\prime }},\\ {m_0}^{\prime }=\frac{y_0-y_M}{{r_0}^{\prime }}=m^{\prime }=\frac{y-y_M}{r^{\prime }},\\ {r_0}^{\prime }=\frac{-z_0}{\cos {\theta }_M},r^{\prime }=\frac{z}{\cos {\theta }_M},\end{array}$$ (6b)

θ_M being the angle indicated in Fig. 1. The linear term of Eq. (2e) now vanishes and we have

$$\begin{array}{l}\mathrm{\Delta }(Q)=\frac{1}{2{\rho }^{\prime }}\{{\left(\xi -x_M\right)}^2+{\left(\eta -y_M\right)}^2\\ -{\left[{l_0}^{\prime }\left(\xi -x_M\right)+{m_0}^{\prime }\left(\eta -y_M\right)\right]}^2\},\end{array}$$ (6c)

$${\rho }^{\prime }=\frac{{r_0}^{\prime }{r}^{\prime }}{{r_0}^{\prime }+r^{\prime }}=\frac{-z{z}_0}{(z-z_0)\cos {\theta }_M}.$$ (6d)

This result can be used in two ways to derive a final result.

3.1.1 Paraxial Approximation

As mentioned in deriving Eq. (4b) above, the direction cosines l₀′ and m₀′ will be small if the points P₀ and P are close to the z-axis of Fig. 1. To a first-order approximation in θ_M we have l′₀², m′₀² « 1, so that the third term of Eq. (6c) can be omitted and Eq. (3c) leads directly to

$$\begin{array}{l}{\alpha }_{\mathrm{F}}(P)\approx -\frac{\mathrm{i}k}{2\mathrm{\pi }{\rho }^{\prime }}{\displaystyle {\int }_{-w}^w\mathrm{d}\xi {\mathrm{e}}^{\mathrm{i}k{\left(\xi -x_{\mathrm{M}}\right)}^2/2{\rho }^{\prime }}{\displaystyle {\int }_{-h}^h\mathrm{d}\eta {\mathrm{e}}^{\mathrm{i}k{\left(\eta -y_{\mathrm{M}}\right)}^2/2{\rho }^{\prime }}}}\\ =-\frac{\mathrm{i}}{2}[F(s_{+})-F(s_{-})][F(t_{+})-F(t_{-})],\end{array}$$ (7a)

where

$$s_{\pm }=\sqrt{\frac{k}{\mathrm{\pi }{\rho }^{\prime }}}(\pm w-x_M),t_{\pm }=\sqrt{\frac{k}{\mathrm{\pi }{\rho }^{\prime }}}(\pm h-y_M),$$ (7b)

and

$$F(s)=C(s)+iS(s)={\displaystyle {\int }_0^s\mathrm{d}\sigma }{\mathrm{e}}^{\mathrm{i}\mathrm{\pi }{\sigma }^2/2}$$ (8)

is the complex Fresnel integral.

3.1.2 Coordinate Transformation for Off-Axis Sources

According to textbooks, a rotation of coordinates may be necessary when the direction cosines l₀′ and m₀′ are too large to justify the paraxial approximation of Sec. 3.1.1, a rotation of coordinates may be necessary. The usual recommendation [3, 5] is to place the new x′-axis along the projection of the line P₀P onto the aperture plane. This gives m₀′ = 0, so that α_F(P) does indeed assume the form stipulated by Eq. (5). However, the x′- and y′-axes so defined are not parallel to the edges of the aperture, and consequently the two integrals are not separable because the limits of the ξ-integral depend on η, and vice versa.

To overcome this difficulty, a different transformation is attempted in the following: The z′-axis is placed in the direction of the unit vector along the line P₀P, so that l₀′ = m₀′ = 0 and Eq. (5) is again satisfied. The x′-axis is chosen so that its projection onto the aperture plane is parallel to the ξ-direction. The y′-axis is defined in the usual manner as y′ = z′ × x′. Thus,

$$\left(\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ z^{\prime }\end{array}\right)=\left(\begin{array}{ccc}1/W& 0& -\cos {\varphi }_M\tan {\theta }_M/W\\ -\sin {\varphi }_M\cos {\varphi }_M\sin {\theta }_M\tan {\theta }_M/W& W\cos {\theta }_M& -\sin {\varphi }_M\sin {\theta }_M/W\\ \cos {\varphi }_M\sin {\theta }_M& \sin {\varphi }_M\sin {\theta }_M& \cos {\theta }_M\end{array}\right)\left(\begin{array}{c}x-x_M\\ y-y_M\\ z\end{array}\right)$$ (9a)

where ϕ_M and θ_M are the longitudes and colatitudes of P₀ and P with respect to M,

$$\tan {\varphi }_M=\frac{y-y_0}{x-x_0},\tan {\theta }_M=\frac{\sqrt{{\left(x-x_0\right)}^2+{\left(y-y_0\right)}^2}}{z-z_0},$$ (9b)

and

$$W=\sqrt{1+{\cos }^2{\varphi }_M{\tan }^2{\theta }_M}.$$ (9c)

Accordingly, for z = 0,

$$\begin{array}{c}{\xi }^{\prime }=\frac{1}{W}(\xi -x_M),\\ {\eta }^{\prime }=W\cos {\theta }_M[\frac{-\sin {\varphi }_M\cos {\varphi }_M{\tan }^2{\theta }_M}{W^2}(\xi -x_M)\\ +(\eta -y_M)].\end{array}$$ (9d)

Equation (9d) shows that η′ is still not independent of ξ. This was to be expected as it is not possible to rotate the z-axis and have orthogonal x- and y-axes which are both aligned with the aperture edges. Accordingly, the separation of integration limits is not complete unless the first term in the above expression for η′ is omitted—a first-order approximation in θ_M. Then,

$$\begin{array}{c}\xi =\frac{1}{W}(\xi -x_M),{\eta }^{\prime }=W\cos {\theta }_M(\eta -y_M),\\ \mathrm{d}{\xi }^{\prime }\mathrm{d}{\eta }^{\prime }=\cos {\theta }_M\mathrm{d}\xi \mathrm{d}\eta .\end{array}$$ (10a)

and

$$\begin{array}{c}{\alpha }_{\mathrm{F}}(P)=\\ -\frac{\mathrm{i}k\cos {\theta }_M}{2\mathrm{\pi }{\rho }^{\prime }}{\displaystyle {\int }_{-w}^w\mathrm{d}\xi {\mathrm{e}}^{\mathrm{i}k{\left(\xi -x_M\right)}^2/2W^2{{\rho }^{\prime }}^2}}{\displaystyle {\int }_{-h}^h\mathrm{d}\eta {\mathrm{e}}^{\mathrm{i}k{W}^2{\cos }^2\theta {\left(\eta -y_M\right)}^2/2{{\rho }^{\prime }}^2}}\\ =-\frac{\mathrm{i}}{2}[F(s_{+}/W)-F(s_{-}/W)][F(W\cos {\theta }_M{t}_{+})\\ -F(W\cos {\theta }_M{t}_{-})]\end{array}$$ (10b)

where s₊ and s₋ are the same as in Eq. (7b), above. It should be noted that this result differs from the paraxial approximation only by the factors 1/W and W cosθ_M in the arguments of the Fresnel integrals. Within the above approximation for η′ these factors are equal to unity, and thus Eq. (10b) appears to be no improvement over the paraxial approximation Eq. (7b) of Sec. 3.1.1. It follows that, for rectangular apertures, the coordinate transformations recommended in Refs. [3] and [5] are superfluous. To higher than first order in θ_M, the ξ- and η-integrals remain inseparable and a closed solution for α_F(P) is not possible.

3.1.3 Evaluation of Fresnel Cosine and Sine Integrals

The use of Eqs. (7a) and (10b) is straightforward. An example is given in Sec. 3.2, below. It should be remembered that the variation of α_F (P) with P is implicit in Eqs. (7b) and (10b), in that x_M, y_M, ϕ_M and θ_M depend on the location of P. It should also be borne in mind that the point M of Fig. 1 will be outside the aperture when P lies in the geometric shadow. This can lead to values of ξ and η larger than assumed in Eq. (3a). For this reason, the computation of α_F (P) based on Eq. (7b) or (10b) must not be carried too far into the shadow region.

The only problem that may be encountered on a personal computer is that the Fresnel cosine and sine integrals defined by Eq. (8) are not usually included in standard software packages. For modest accuracy requirements, they can be computed from the equations quoted in Ref. [9],

$$\begin{array}{c}C(s)=\frac{1}{2}+f(s)\sin \left(\frac{\mathrm{\pi }{s}^2}{2}\right)-g(s)\cos \left(\frac{\mathrm{\pi }{s}^2}{2}\right),\\ C(-s)=-C(s),\end{array}$$ (11a)

$$\begin{array}{c}S(s)=\frac{1}{2}-f(s)\cos \left(\frac{\mathrm{\pi }{s}^2}{2}\right)-g(s)\sin \left(\frac{\mathrm{\pi }{s}^2}{2}\right),\\ S(-s)=-S(s),\end{array}$$ (11b)

$$f(s)=\frac{1+0.926s}{2+1.792s+3.104s^2}+ϵ(s),s\ge 0,$$ (11c)

$$g(s)=\frac{1}{2+4.142s+3.492s^2+6.67s^3}+ϵ(s),s\ge 0,$$ (11d)

where |ϵ (s)| ≤ 2 × 10⁻³. Accordingly, the following simple algorithm may be used:

1. Define s.

2. Let s′ = |s|.

3. Calculate f(s′), g(s′) from Eq. (11c,d).

4. Calculate C(s′), S(s′) from Eq. (11a,b).

5. If s < 0 let C(s) = − C(s′), S(s) = − S(s′). Else, let C(s) = C(s′), S(s) = S(s′).

If better accuracy is desired, this algorithm can be improved by using the method described in Ref. [10]. Alternatively, software for computing C(s) and S(s) in Fortran or C can be downloaded [11, 12].

3.2 Application to Slits (Fig. 2)

The rectangular aperture discussed so far is transformed into a slit of width 2w on setting h = ∞ in Eq. (7b).5 It may then also be assumed that the source is a long luminous line which is parallel to the slit and passes through the point P₀ in Fig. 1, so that it will suffice to compute the diffraction pattern in the xz-plane shown in Fig. 2. With these assumptions we have t± = ± ∞, so that F(t_±) = ± ½ (1 + i), [F(t_±) − F(t₋)] = 1 + i, and Eq. (10b) is reduced to

$$\begin{array}{c}{\alpha }_{\mathrm{F}}(P)=\frac{1-i}{2}[F(s_{+})-F(s_{-})]\\ =\frac{1-\mathrm{i}}{2}\{[C(s_{+})-C(s_{-})]+\mathrm{i}[S(s_{+})-S(s_{-})]\},\end{array}$$ (12a)

with s as defined by Eq. (7b) but assuming y₀ = y = y_M = 0 so that Eq. (9b) is simplified to

$${\theta }_M=\arctan \frac{x_0-x_M}{z_0},x=x_M+z\tan {\theta }_M.$$ (12b)

As the diffraction pattern is centered at and symmetrical about the geometrical source image C shown in Fig. 2, where

$$x_M=0,{\theta }_M={\theta }_C=\arctan \frac{x_0}{z_0},x=x_C=\frac{x_{0}z}{z_0},$$ (12c)

it will also suffice to compute it for positive value values of x_M, only. The computation is typically carried to a maximum value of θ_M beyond the shadow boundary S, the latter being given by

$$x_M=w,{\theta }_M={\theta }_S=\arctan \frac{x_0-w}{z_0},x=x_S=w+z\tan {\theta }_S.$$ (12d)

Accordingly, the following procedure may be used to evaluate the dependence of α_F(P) on x:

1. Define a maximum (x_M)_max and a step size Δx_M for x_M.

2. Let x_M = 0.

3. Compute θ_M and x from Eq. (12b) and s_± from Eq. (7b). Use the algorithm of Sec. 3.1 to find C(s_±) and S(s_±). Compute α_F(P) from Eq. (12a).

4. Let x_M = x_M + Δx_M. If x_M < (x_M)_max, go to Step 3. Else, stop.

A typical diffraction pattern computed in this manner is shown in Fig. 3. The numerical parameters chosen for this particular example are listed in the figure caption and were taken from an experiment described by Fresnel.6

Fig. 3.

Relative irradiance, |α_F(u, v)|² vs v/u, for a slit. w = 1 mm, z₀ = − 2.507 m, z = 1.140 m, λ = 639 nm.

4. Circular Aperture (Fig. 4)

4.1 General Theory

In evaluating the Fresnel diffraction integral Eqs. (3a–c) for a circular aperture with diameter 2a it is convenient to use spherical coordinates centered at the aperture center O, so that

$$\begin{array}{c}{P}_0=(x_0,y_0,z_0)=r_0(\cos {\varphi }_0\sin {\theta }_0,\sin {\varphi }_0\sin {\theta }_0,\cos {\theta }_0)\\ =-r_0(l_0,m_0,n_0),\end{array}$$ (13a)

$$Q=(\xi ,\eta ,0)=q(\cos \chi ,\sin \chi ,0),$$ (13b)

$$\begin{array}{c}P=(x,y,z)=r(\cos \varphi \sin \theta ,\sin \varphi \sin \theta ,\cos \theta )\\ =r(l,m,n)\end{array}$$ (13c)

In these coordinates, the path difference Δ(Q) defined by Eq. (2e) can be evaluated as follows.

Let C = r(l₀, m₀, n₀) be the geometrical image of P₀ at the distance r from the aperture center, so that the position of P relative to C will be given by the vector CP = r(l − l₀, m − m₀, n − n₀). Let F be the foot of the perpendicular from C onto the xy-plane, let c = FP, and define

$$\mathit{FP}=r(l-l_0,m-m_0,0)=c(\cos \gamma ,\sin \gamma ,0),$$ (14a)

where

$$\begin{array}{c}c=r\sqrt{{\left(l-l_0\right)}^2+{\left(m-m_0\right)}^2}\\ =r\sqrt{{\sin }^2\theta +{\sin }^2{\theta }_0+2\sin \theta \sin {\theta }_0\cos (\varphi -\varphi ),}\end{array}$$ (14b)

$$\tan \gamma =\frac{m-m_0}{l-l_0}=\frac{\sin \varphi \sin \theta +\sin {\varphi }_0\sin {\theta }_0}{\cos \varphi \sin \theta +\cos {\varphi }_0\sin {\varphi }_0}.$$ (14c)

Accordingly, the linear term of Eq. (2e) can be expressed in the form

$$\begin{array}{c}(l-l_0)\xi +(m-m_0)\eta =\frac{1}{r}\mathit{FP}\cdot \mathit{OQ}\\ =\frac{qc}{r}(\cos \chi \cos \gamma +\sin \chi \sin \gamma )=\frac{qc}{r}\cos (\chi -\gamma ).\end{array}$$ (14d)

In the quadratic term of Eq. (2e), we have

$$\begin{array}{c}\left({\xi }^2+{\eta }^2\right)-{\left(l_0\xi +m_0\eta \right)}^2\\ =q^2\left[1-{\sin }^2{\theta }_0{\left(\cos {\varphi }_0\cos \chi +\sin {\varphi }_0\sin \chi \right)}^2\right]\\ =q^2\left[1-{\sin }^2{\theta }_0{\cos }^2\left(\chi -{\varphi }_0\right)\right]\end{array}$$ (15a)

and, likewise,

$$\left({\xi }^2+{\eta }^2\right)-{\left(l\xi +m\eta \right)}^2=q^2\left[1-{\sin }^2\theta {\cos }^2\left(\chi -\varphi \right)\right].$$ (15b)

In the following, it will be assumed that the points P₀ and P are close to the z-axis so that sin²θ₀ and sin²θ are negligibly small compared to unity. In this paraxial approximation one obtains

$$\begin{array}{c}\frac{1}{2r_0}\left[\left({\xi }^2+{\eta }^2\right)-{\left(l_0\xi +m_0\eta \right)}^2\right]\\ +\frac{1}{2r}\left[\left({\xi }^2+{\eta }^2\right)-{\left(l\xi +m\eta \right)}^2\right]\approx \frac{r_0+r}{2r_{0}r}{q}^2.\end{array}$$ (15c)

On substitution of Eqs. (14d) and (15c) into Eq. (2e) we have

$$\mathrm{\Delta }(Q)=-\frac{qc}{r}\cos (\chi -\gamma )+\frac{r_0+r}{2r_{0}r}{q}^2,$$ (16a)

and hence Eq. (3c) is reduced to

$$\begin{array}{c}{\alpha }_{\mathrm{F}}(P)=-\frac{\mathrm{i}k(r_0+r)}{2\mathrm{\pi }{r}_{0}r}{\displaystyle {\int }_0^a\mathrm{d}qq}{\mathrm{e}}^{\mathrm{i}k\frac{r_0+r}{2r_{0}r}{q}^2}{\displaystyle {\int }_0^{2\mathrm{\pi }}d\chi {\mathrm{e}}^{-\mathrm{i}k\frac{qc}{r}\cos (\chi -\gamma )}}\\ =-\frac{\mathrm{i}k(r_0+r)}{r_{0}r}{\displaystyle {\int }_0^a\mathrm{d}qq}{J}_0\left(k\frac{qc}{r}\right){\mathrm{e}}^{\mathrm{i}k\frac{r_0+r}{2r_{0}r}{q}^2},\end{array}$$ (16b)

where the integral over χ was evaluated as 2πJ₀(kqc/r) [9]. On substituting

$$\rho =\frac{q}{a},u=\frac{k{a}^2(r_0+r)}{r_{0}r},v=\frac{kac}{r},$$ (17a)

this becomes

$${\alpha }_{\mathrm{F}}(P)=-iu{\displaystyle {\int }_0^1\mathrm{d}\rho \rho J_0(v\rho ){\mathrm{e}}^{\frac{i}{2}u{\rho }^2}}.$$ (17b)

As expected, these equations describe a circular diffraction pattern which is fully determined by a radial variable, c or v. The pattern is centered at the geometrical source image C, defined by c = v = 0, and α_F(P) is constant on any circle about C. The radius of the geometrically illuminated spot at the distance r from the aperture is a (r₀ + r)/r₀, so that in the notation of Eq. (17a) the geometrical shadow boundary is defined by v = u. The parameter u, which relates the aperture radius a to the wavelength λ and the distances r₀ and r,7 can assume widely different values. For example, in the case of a classroom demonstration of Fresnel diffraction, the parameters λ = 500 nm, a = 0.1 mm, r₀ = r = 100 mm are typical and in this case one has u = 0.8p. On the other hand, for limiting apertures used in a radiometer, parameters such as λ = 500 nm, a = 5 mm, r₀ = r = 1 m are typical, and then one has u = 200π. As will be shown later, the diffraction patterns encountered in these different cases are very different. For u → 0 the diffraction pattern approaches the Fraunhofer limit (Airy function), and for u → ∞ it approaches the limit of geometrical optics (rectangle function). (See Sec. 4.2, Figs. 6a–d).

Fig. 6.

Relative irradiances for circular apertures: a, b, c) |α_L(u, v)|² vs v/u for u = 1, 10, and 100 according to Sec. 4.2. d) |α_Schw(u, v)|² vs v/u u for u = 1000 according to Sec. 4.3.

4.2 Lommel’s Solution

Lommel [2] evaluated the integral Eq. (17b) in the form8

$${\displaystyle {\int }_0^1\mathrm{d}\rho \rho }{J}_0(v\rho ){\mathrm{e}}^{\frac{i}{2}u{\rho }^2}=\frac{1}{2}[\mathrm{L}(u,v)+\text{iM}(u,v)],$$ (18a)

so that

$${\alpha }_{\mathrm{F}}(u,v)\equiv {\alpha }_{\mathrm{L}}(u,v)=\frac{u}{2}\mathrm{M}(u,v)-\mathrm{i}\frac{u}{2}\mathrm{L}(u,v).$$ (18b)

In this notation,

$${\left|{\alpha }_{\mathrm{L}}(u,v)\right|}^2=\frac{u^2}{4}[{\mathrm{M}}^2(u,v)+{\mathrm{L}}^2(u,v)]$$ (18c)

is the relative irradiance of the diffracted light and

$${\Phi }_{\mathrm{L}}(u,v)=\arctan \frac{\mathrm{Im}({\alpha }_{\mathrm{L}})}{\mathrm{Re}({\alpha }_{\mathrm{L}})}=-\arctan \frac{\mathrm{L}(u,v)}{\mathrm{M}(u,v)}$$ (18d)

is the phase difference relative to the geometric field.

The functions L(u, v) and M(u, v) appearing in these equations are defined by

$$\begin{array}{c}\frac{u}{2}\mathrm{L}(u,v)=\sin \frac{v^{\mathrm{2}}}{2u}+{\mathrm{V}}_0(u,v)\sin \frac{u}{2}-{\mathrm{V}}_1(u,v)\cos \frac{u}{2}\\ ={\mathrm{U}}_1(u,v)\cos \frac{u}{2}+{\mathrm{U}}_2(u,v)\sin \frac{u}{2},\end{array}$$ (19a)

$$\begin{array}{c}\frac{u}{2}\mathrm{M}(u,v)=\cos \frac{v^{\mathrm{2}}}{2u}+{\mathrm{V}}_0(u,v)\cos \frac{u}{2}-{\mathrm{V}}_1(u,v)\sin \frac{u}{2}\\ ={\mathrm{U}}_1(u,v)\sin \frac{u}{2}-{\mathrm{U}}_2(u,v)\cos \frac{u}{2},\end{array}$$ (19b)

where

$${\mathrm{V}}_0(u,v)={\mathrm{J}}_0(v)-\left(\frac{v}{u}\right){ }^2{\mathrm{J}}_2(v)+\left(\frac{v}{u}\right){ }^4{\mathrm{J}}_4(v)+-\dots ,$$ (20a)

$${\mathrm{V}}_1(u,v)=\left(\frac{v}{u}\right){\mathrm{J}}_1(v)-\left(\frac{v}{u}\right){ }^3{\mathrm{J}}_3(v)+\left(\frac{v}{u}\right){ }^5{\mathrm{J}}_5(v)+-\dots ,$$ (20b)

$${\mathrm{U}}_1(u,v)=\left(\frac{u}{v}\right){\mathrm{J}}_1(v)-\left(\frac{u}{v}\right){ }^3{\mathrm{J}}_3(v)+\left(\frac{u}{v}\right){ }^5{\mathrm{J}}_5(v)+-\dots ,$$ (20c)

$${\mathrm{U}}_2(u,v)=\left(\frac{u}{v}\right){ }^2{\mathrm{J}}_2(v)-\left(\frac{u}{v}\right){ }^4{\mathrm{J}}_4(v)+\left(\frac{u}{v}\right){ }^6{\mathrm{J}}_6(v)+-\dots ,$$ (20d)

are Lommel functions of two variables, J_n(v) being a Bessel function of the first kind and order n.

For checking the accuracy of numerical results, it is useful to note the values of these expressions in special cases:

1. In the limit u → 0, Lommel’s equations simplify to the familiar Airy formula for Fraunhofer diffraction at a circular aperture. In this case, the entire diffraction pattern lies in the geometrical shadow and Eqs. (20c,d) are reduced to

   $$\frac{{\mathrm{U}}_1(u,v)}{u}\to \frac{{\mathrm{J}}_1(v)}{v},\frac{{\mathrm{U}}_2(u,v)}{u}\to 0,{\alpha }_{\mathrm{L}}(P)\to \frac{-\mathrm{i}u{\mathrm{J}}_1(v)}{v},$$ (21)

   so that Airy’s formula, U(P) ∝ J₁(v)/v, is obtained from Eqs. (17a) and (3a,b).
2. For v = 0, we have

   $$\mathrm{Re}[{\alpha }_{\mathrm{L}}(u,0)]=\left(1-\cos \frac{u}{2}\right),\mathrm{Im}[{\alpha }_{\mathrm{L}}(u,0)]=-\sin \frac{u}{2},$$ (22a)

   $${\left|{\alpha }_{\mathrm{L}}(u,0)\right|}^2=2\left(1-\cos \frac{u}{2}\right),$$ (22b)

   $${\Phi }_{\mathrm{L}}(u,0)=-\arctan \left[\frac{\sin (u/2)}{1-\cos (u/2)}\right].$$ (22c)
3. For v = u, the well-known relations [9]

   $$\frac{1}{2}\cos u=\frac{1}{2}{\mathrm{J}}_0(u)-{\mathrm{J}}_2(u)+{\mathrm{J}}_4(u)\dots ,$$ (23a)

   $$\frac{1}{2}\sin u={\mathrm{J}}_1(u)-{\mathrm{J}}_3(u)+{\mathrm{J}}_5(u)\dots ,$$ (23b)

   may be used to show that

   $${\mathrm{V}}_0(u,u)=\frac{1}{2}[{\mathrm{J}}_0(u)+\cos u],{\mathrm{V}}_1(u,u)=\frac{1}{2}\sin u,$$ (23c)

   $${\mathrm{U}}_1(u,u)=\frac{1}{2}\sin u,{\mathrm{U}}_2(u,u)=\frac{1}{2}[{\mathrm{J}}_0(u)-\cos u].$$ (23d)

   Accordingly,

   $$\mathrm{Re}[{\alpha }_{\mathrm{L}}(u,u)]=\frac{1}{2}[1-{\mathrm{J}}_0(u)]\cos \frac{u}{2},$$

   $$\mathrm{Im}[{\alpha }_{\mathrm{L}}(u,u)]=-\frac{1}{2}[1+{\mathrm{J}}_0(u)]\sin \frac{u}{2},$$ (23e)

   $${\left|{\alpha }_{\mathrm{L}}(u,u)\right|}^2=\frac{1}{4}[1-2{\mathrm{J}}_0(u)\cos u+{\mathrm{J}}_0^2(u)].$$ (23f)

   $${\Phi }_{\mathrm{L}}(u,u)=-\arctan \left[\frac{1+{\mathrm{J}}_0(u)}{1-{\mathrm{J}}_0(u)}\tan \frac{u}{2}\right]$$ (23g)

The use of Lommel’s equations for numerical computations is straightforward, provided that accurate values of the Bessel functions J_n(v) required for Eqs. (20a–d) are available and the convergence behavior of these equations is taken into consideration.

When v/u or u/v are small, these expansions will converge on account of the monotonic decrease of (v/u)ⁿ or (u/v)ⁿ, provided of course that L and M are evaluated in terms of the Lommel functions V₀ and V₁ when v < u and in terms of U₁ and U₂ when v > u.

When v/u or u/v are close to unity, Eqs. (20a–d) will converge on account of the relation J_n(v) → 0 when n → ∞. The manner in which this limit is approached is illustrated in Fig. 5. As n increases, J_n(v) exhibits an oscillatory behavior before vanishing after passing through a pronounced final maximum at or below n = v. In this case, L and M can be evaluated in terms V₀, V₁ or U₁,U₂, although for consistency it is better to use the former method when v < u and the latter method when v > u.

Fig. 5.

Dependence of Bessel functions J_n(v) on n.

It follows that, for computations of the Lommel functions to m decimals, the expansions Eqs. (20a–d) can be truncated when either of the conditions,

$${\left(\frac{v}{u}\right)}^n\text{or}{\left(\frac{u}{y}\right)}^n<\frac{1}{2}{10}^{-m},$$ (24a)

$$n\ge v{\text{and J}}_n(v)<\frac{1}{2}{10}^{-m},$$ (24b)

are satisfied.

The numerical results presented in this paper were obtained on a personal computer, using standard spreadsheet software9 (a 133 Mhz Pentium computer and Microsoft Excel 7.0). It was found that this software provides accurate values of the Bessel functions J_n(v) needed for Eqs. (20a–d) without problems, but that the large number of them required to satisfy Eq. (24b) impeded the speed of program execution when u is large and v ≈ u. In addition, the capabilities of the personal computer were overtaxed by the fact that the diffraction patterns for large values of u are highly structured (see Figs. 6a–d), so that a very large number of data points had to be computed. For these reasons, Eqs. (18a) to (20d) were used in this work only for u ≤ 300 while for larger values of the approximation of Sec. 4.3, below, was used. It should be emphasized that this limitation is unnecessary for larger computers. When sufficient computing power is available, the Lommel functions for large values of u can be evaluated efficiently by iterative use of recurrence relations for Bessel functions, beginning at the required large orders and iterating towards J₀(v) from above, as mentioned by Shirley and Datla [13]. Under these conditions, the fine structure of the diffraction pattern poses no difficulties.

The algorithm used in this work for u ≤ 300 was as follows.

1. Define the value of u, a maximum value v_max, a step size Δv, and the desired decimal accuracy 10^−m.

2. Let v = 0. Compute α_L(u, 0) from Eq. (22a).

3. Let v = v + Δv. Compute (u/2) V₀(u, v), (u/2)·V₁(u, v) from Eq. (20a,b), terminating when Eq. (24a or b) are satisfied. Compute L(u, v), M(u, v) from Eq. (19a,b) and α_L(u, v) from Eq. (18b).

4. If v < u, go to Step 3.

5. Compute α_L(u, u) from Eq. (23e).

6. Let v = v + Δv. Compute (u/2) U₁(u, v), (u/2) U₂(u, v) from Eq. (20c,d), terminating when Eq. (24a or b) are satisfied. Compute L(u, v), M(u, v) from Eq. (19a,b) and α_L(u, v) from Eq. (18b).

7. If v < v_max, go to Step 6. Else, stop.

The relative irradiances |α_L(u, v)|² computed in this manner for u = 1, 10, and 100 are plotted in Figs. 6a–c.

4.3 Schwarzschilds’s Approximation

The above-mentioned computational problems encountered with Lommel’s solution when u is large can be avoided by using an asymptotic approximation for α_F(P) derived by Schwarzschild [8] in a paper on diffraction effects in defocused telescopes. As this paper is no longer readily available, its contents will be outlined here.

Schwarzschild considered the integral

$$\begin{array}{l}W=\frac{\mathrm{i}u}{2\mathrm{\pi }}{\displaystyle {\int }_0^1\mathrm{d}\rho \rho }{\displaystyle {\int }_0^{2\mathrm{\pi }}\mathrm{d}\chi {\mathrm{e}}^{-\mathrm{i}(u{\rho }^2/2-v\rho \cos \chi +v^2/2u)}}\\ =\frac{\mathrm{i}u}{2\mathrm{\pi }}\left[{\displaystyle {\int }_0^{\infty }\mathrm{d}\rho \dots -{\displaystyle {\int }_1^{\infty }\mathrm{d}\rho }\dots }\right]=W_1-W_2,\end{array}$$ (25a)

so that his approximation of Eq. (17b) is given by10

$${\alpha }_{\mathrm{L}}(u,v)\approx {\alpha }_{\text{Schw}}(u,v)={\mathrm{e}}^{\frac{\mathrm{i}{y}^2}{2u}}W*.$$ (25b)

The integral W₁ is readily shown to be equal to

$$W_1=1,$$ (26a)

and by evaluating the x -integral of W₂ as in Eq. (16b) and then substituting the asymptotic expression [9]

$${\mathrm{J}}_0(v\rho )=\sqrt{\frac{2}{\mathrm{\pi }\rho v}}\cos \left(v\rho -\frac{\mathrm{\pi }}{4}\right),v\rho >>1$$ (26b)

one obtains

$$\begin{array}{c}{W}_2\approx \frac{1}{\sqrt{2\mathrm{\pi }v}}[{\mathrm{e}}^{\frac{\mathrm{i}\mathrm{\pi }}{4}}\mathrm{i}u{\displaystyle {\int }_1^{\infty }\mathrm{d}\rho \sqrt{\rho }{\mathrm{e}}^{-\frac{\mathrm{i}u}{2}{\left(\rho +v/u\right)}^2}}\\ +{\mathrm{e}}^{-\frac{\mathrm{i}\mathrm{\pi }}{4}}\mathrm{i}u{\displaystyle {\int }_1^{\infty }\mathrm{d}\rho }\sqrt{\rho }{\mathrm{e}}^{-\frac{\mathrm{i}u}{2}{\left(\rho -v/u\right)}^2}].\end{array}$$ (26c)

Letting

$$f=\frac{\sqrt{\rho }}{\rho +v/u},g=-{\mathrm{e}}^{-\frac{\mathrm{i}u}{2}{\left(\rho +v/u\right)}^2},$$ (27a)

the first integral in Eq. (26c) can be expressed in the form ∫f dg so that

$$\begin{array}{l}\mathrm{i}u{\displaystyle {\int }_1^{\infty }\mathrm{d}\rho }\sqrt{\rho }{\mathrm{e}}^{-\frac{\mathrm{i}u}{2}{\left(\rho +v/u\right)}^2}=\frac{{\mathrm{e}}^{-\frac{\mathrm{i}{\left(u+v\right)}^2}{2u}}}{1+v/u}\\ +{\displaystyle {\int }_1^{\infty }\mathrm{d}\left(\frac{\sqrt{\rho }}{\rho +v/u}\right){\mathrm{e}}^{-\frac{\mathrm{i}u}{2}{\left(\rho +v/u\right)}^2}}.\end{array}$$ (27b)

The second integral in Eq. (26b) can be written as

$$\begin{array}{c}\mathrm{i}u{\displaystyle {\int }_1^{\infty }\mathrm{d}\rho }\sqrt{\rho }{\mathrm{e}}^{-\frac{\mathrm{i}u}{2}{\left(\rho -v/u\right)}^2}\\ =\mathrm{i}u{\displaystyle {\int }_1^{\infty }\mathrm{d}\rho }(\sqrt{\rho }-\sqrt{v/u}){\mathrm{e}}^{-\frac{\mathrm{i}u}{2}{\left(\rho -v/u\right)}^2}\\ +\mathrm{i}\sqrt{uv}{\displaystyle {\int }_1^{\infty }\mathrm{d}\rho {\mathrm{e}}^{-\frac{\mathrm{i}u}{2}{\left(\rho -v/u\right)}^2}}\end{array}$$ (28a)

where the first term can again be evaluated by partial integration, using

$$f=\frac{\sqrt{\rho }-\sqrt{v/u}}{\rho -v/u}=\frac{1}{\sqrt{\rho }+\sqrt{v/u}},$$ (28b)

and the second term is a complex Fresnel integral [Eq. (8)]. In this manner, Schwarzschild found

$$\begin{array}{c}\mathrm{i}u{\displaystyle {\int }_1^{\infty }\mathrm{d}\rho }\sqrt{\rho }{\mathrm{e}}^{-\frac{\mathrm{i}u}{2}{\left(\rho -v/u\right)}^2}\\ =\mathrm{i}\sqrt{\mathrm{\pi }v}\{F*(\infty )-F*[\sqrt{u/\mathrm{\pi }}(1-v/u)]\}\\ +\frac{{\mathrm{e}}^{-\frac{\mathrm{i}{\left(u-v\right)}^2}{2u}}}{1+\sqrt{v/u}}+{\displaystyle {\int }_1^{\infty }\mathrm{d}\left(\frac{1}{\sqrt{\rho }+\sqrt{v/u}}\right){\mathrm{e}}^{-\frac{\mathrm{i}u}{2}{\left(\rho -v/u\right)}^2}.}\end{array}$$ (28c)

He noted that, by further partial integrations, the resulting expression for W₂ would become an asymptotic expansion in negative powers of u but that there was no point in taking the trouble as the last terms in Eqs. (27b) and (28c) are already negligibly small for practical purposes. Therefore,

$$\begin{array}{c}W=1-W_2\approx \frac{1}{2}+\frac{\mathrm{i}}{\sqrt{2}}{\mathrm{e}}^{-\frac{\mathrm{i}\mathrm{\pi }}{4}}F*[\sqrt{u/\mathrm{\pi }}(1-v/u)]\\ -\frac{1}{\sqrt{2\mathrm{\pi }v}}\left[\frac{{\mathrm{e}}^{-\frac{\mathrm{i}{\left(u+v\right)}^2}{2u}+\frac{\mathrm{i}\mathrm{\pi }}{4}}}{1+v/u}+\frac{{\mathrm{e}}^{-\frac{\mathrm{i}{\left(u-v\right)}^2}{2u}-\frac{\mathrm{i}\mathrm{\pi }}{4}}}{1+\sqrt{v/u}}\right],u,v>>1.\end{array}$$ (29)

Schwarzschild estimated that this expression is accurate to 0.005 if u = 100, v/u > 0.2; or u = 300, v/u > 30.

When put in the form of Eq. (25b), Schwarzschild’s approximation becomes

$$\begin{array}{l}{\alpha }_{\mathrm{F}}(u,v)\approx {\alpha }_{\text{Schw}}(u,v)=\frac{1}{2}{\mathrm{e}}^{-\mathrm{i}\delta }-\frac{\mathrm{i}}{\sqrt{2}}{\mathrm{e}}^{-\mathrm{i}(\delta -\mathrm{\pi }/4)}\mathrm{F}(s)\\ -\frac{1}{\sqrt{2\mathrm{\pi }v}}\left[\frac{{\mathrm{e}}^{\mathrm{i}({\beta }_{+}-\mathrm{\pi }/4)}}{1+v/u}+\frac{{\mathrm{e}}^{\mathrm{i}({\beta }_{-}+\mathrm{\pi }/4)}}{1+\sqrt{v/u}}\right],u,v>>1,\end{array}$$ (30a)

$$\begin{array}{c}\mathrm{Re}[{\alpha }_{\text{Schw}}(u,v)]=\frac{1}{2}\cos \delta -\frac{1}{\sqrt{2}}[C(s)\sin (\delta -\mathrm{\pi }/4)\\ -S(s)\cos (\delta -\mathrm{\pi }/4)]-\frac{1}{\sqrt{2\mathrm{\pi }v}}[\frac{\cos ({\beta }_{+}-\mathrm{\pi }/4)}{1+v/u}\\ +\frac{\cos ({\beta }_{-}+\mathrm{\pi }/4)}{1+\sqrt{v/u}}],u,v>>1,\end{array}$$ (30b)

$$\begin{array}{c}\mathrm{Im}[{\alpha }_{\text{Schw}}(u,v)]=-\frac{1}{2}\sin \delta -\frac{1}{\sqrt{2}}[C(s)\cos (\delta -\mathrm{\pi }/4)\\ +S(s)\sin (\delta -\mathrm{\pi }/4)]-\frac{1}{\sqrt{2\mathrm{\pi }v}}[\frac{\sin ({\beta }_{+}-\mathrm{\pi }/4)}{1+v/u}\\ +\frac{\sin ({\beta }_{-}+\mathrm{\pi }/4)}{1+\sqrt{v/u}}],u,v>>1..,\end{array}$$ (30c)

where

$$\delta =\frac{v^2}{2u},{\beta }_{\pm }=\frac{u}{2}\pm v,s=\sqrt{u/\mathrm{\pi }}(1-v/u).$$ (30d)

The use of these expressions on a computer is simple. The only caveat is that they are not valid for small values of v/u so that Lommel’s equations must still be used below a suitably chosen minimum value v = v_min. The choice of v_min can be based on the following table, which was obtained by computing the residuals Δ = |α_Schw|² − |α_L|² for selected values of u.

$$\begin{array}{ccc}u& \mathrm{\Delta }\le 0.01& \mathrm{\Delta }\le 0.001\\ 30& \text{if}v/u\ge 0.23& \text{if}v/u\ge 0.7\\ 100& \text{if}v/u\ge 0.06& \text{if}v/u\ge 0.27\end{array}$$

For such small values of v/u, only a few terms of Eqs. (20a,b) are sufficient to obtain α_L with comparable accuracy. Thus, the following procedure will provide the entire diffraction pattern.

1. Define u, v_min, v_max, and Δv.

2. Let v = 0. Compute α_L(0, v) from Eq, (22a).

3. Let v = v + Δv. Compute (u/2)·V₀(u, v), (u/2)·V₁(u, v) from Eqs. (20a,b), terminating after the third terms. Compute L(u, v), M(u, v) from Eqs. (19a,b) and α_L(u, v) from Eq. (18b).

4. If v < v_min, go to Step 3.

5. Let v = v + Δv. Compute δ, β_±, s from Eq. (30d). Use the algorithm of Sec. 3.1 to find C(s), S(s). Compute α_Schw(u, v) from Eqs. (30b,c).

6. If v < v_max, go to Step 5. Else, stop.

The values of |α_Schw(u, v)|² computed by this algorithm for u = 1000 are shown in Fig. 6d. They were found to be in excellent agreement with the values of |α_L(u, v)|² obtained from Lommel’s solution.11

It should be emphasized that Schwarzschild’s approximation is different from a superficially similar but significantly less accurate asymptotic approximation of the diffraction integral (17b) cited by Focke [7] and used by Blevin [14], Steel, De, and Bell [16], and Boivin [16] in their work on diffraction errors in radiometry. The respective accuracies of the Schwarzschild and Focke approximations can be assessed by comparing the relative irradiances at the shadow boundary,12

$$\begin{array}{c}{\left|{\alpha }_{\text{Schw}}(u,u)\right|}^2=\frac{1}{4}[1-\sqrt{\frac{8}{\mathrm{\pi }u}}\cos u\cos \left(u-\frac{\mathrm{\pi }}{4}\right)\\ +\frac{2}{\mathrm{\pi }u}{\cos }^2\left(u-\frac{\mathrm{\pi }}{4}\right)],\end{array}$$ (31a)

$${\left|{\alpha }_{\text{Focke}}(u,u)\right|}^2=\frac{1}{4}\left[1-\frac{1}{\sqrt{\mathrm{\pi }u}}(\cos 2u+\sin 2u)+\frac{1}{2\mathrm{\pi }u}\right],$$ (31b)

to the exact value |α_L(u, u)|² given by Eq. (23f). From the ratios plotted in Fig. 7 it follows that the Schwarzschild values are accurate to 0.1 % for u > 10, while for u = 100 the Focke values are still off by 6 %.

Fig. 7.

Irradiance ratios, |α_Schw(u, u)|²/|α_L(u, u)|² and |α_Focke(u, u)|²/|α_L(u, u)|², vs u.

Biography

About the author: Klaus D. Mielenz is a physicist and retired Chief of the Radiometric Physics Division of NIST. The National Institute of Standards and Technology is an agency of the Technology Administration, U.S. Department of Commerce.