Fresnel Diffraction by Convex Bodies

V. A. Fock

The approximate method of calculating diffraction based on Huygens’ principle makes it possible, as is known, to find the field of a wave bending around a thin opaque screen; this field is expressed in terms of Fresnel integrals.

In the case, however, when the body being bypassed has finite curvature (the radius of curvature is large in comparison with the wavelength), the question of approximate formulas for the field in the region of the geometrical boundary of the shadow at a sufficiently large distance from the body has remained open; in particular, it had not been clarified whether, in this case, those expressions for the field in terms of Fresnel integrals which can be constructed by analogy with the case of an infinitely thin screen are applicable.

In the present work we shall show, using diffraction by a sphere as an example, that for bodies of finite curvature as well the principal term in the expression for the field behind the body is expressed in terms of Fresnel integrals. This term does not depend (as in the case of ordinary Fresnel diffraction) on the material of the body around which the wave bends. But an additional term is added to the principal term, constituting, as it were, a background on which the fringes of Fresnel diffraction are situated, and this additional term (and hence also the background) already depends on the electrical properties of the body around which the wave bends.

1. Initial Formulas for the Attenuation Factor

We shall proceed from the diffraction formulas derived in our work1. We must here summarize the principal results of that work. The field of a point source (dipole) situated at some distance from the surface of a sphere is expressed in terms of two functions \(U\) and \(W\), which represent solutions of the wave equation:

\[ \Delta U + k^2 U = 0 \tag{1.01} \]

and have a point singularity of the form

\[ U=\frac{e^{ikR}}{R}+U^0, \tag{1.02} \]

where \(R\) is the distance from the source, and \(U^0\) remains finite as \(kR\to 0\).

The equations determining \(U\) and \(W\) differ from one another in the form of the boundary conditions, which we shall not write out here.

Let \(r,\vartheta,\varphi\) be spherical coordinates with origin at the center of the sphere and with polar axis passing through the dipole. The quantity \(s=a\vartheta\), where \(a\) is the radius of the sphere, will be the distance from the source to the point of observation, measured along an arc of the sphere. We denote the height of the source above the surface of the sphere by \(h_1\), and the height of the point of observation by \(h_2\). Introduce the parameter

\[ m=\sqrt[3]{\frac{ka}{2}}, \tag{1.03} \]

which we shall assume to be large, and put

\[ x=\sqrt[3]{\frac{k}{2a^2}}\,s = m\frac{s}{a}=m\vartheta; \tag{1.04} \]

\[ y_1=\frac{kh_1}{m};\qquad y_2=\frac{kh_2}{m}. \tag{1.05} \]

We denote the complex dielectric constant of the material of the sphere by \(\eta\), and shall assume that \(|\eta|\gg 1\). Finally, put

\[ q=\frac{im}{\sqrt{\eta+1}};\qquad q_1=im\sqrt{\eta-1}. \tag{1.06} \]

In our work it is shown that near the surface of the sphere (at distances small in comparison with its radius) the functions \(U\) and \(W\) are expressed in terms of the attenuation factor \(V\) by the formulas

\[ U=\frac{e^{iks}}{\sqrt{s a\sin \dfrac{s}{a}}}\cdot V(x,y_1,y_2,q), \tag{1.07} \]

\[ W=\frac{e^{iks}}{\sqrt{s a\sin \dfrac{s}{a}}}\cdot V(x,y_1,y_2,q_1). \tag{1.08} \]

For \(y_1<y_2\) the attenuation factor \(V\) can be represented in the form of the contour integral

\[ V(x,y_1,y_2,q)=e^{-i\frac{\pi}{4}}\sqrt{\frac{x}{\pi}} \int_C e^{ixt}F(t,y_1,y_2,q)\,dt, \tag{1.09} \]

where the function \(F\) can be written in the form:

\[ F=w_1(t-y_2)\left\{v(t-y_1)-\frac{v'(t)-qv(t)}{w_1'(t)-qw_1(t)}\,w_1(t-y_1)\right\}, \tag{1.10} \]

or else in the form:

\[ F=\frac{i}{2}w_1(t-y_2)\left\{w_2(t-y_1)-\frac{w_2'(t)-qw_2(t)}{w_1'(t)-qw_1(t)}\,w_1(t-y_1)\right\}. \tag{1.11} \]

Here \(w_1(t)\) and \(w_2(t)\) are complex Airy functions representing solutions of the differential equation

\[ w''(t)=tw(t) \tag{1.12} \]

and having, for large negative \(t\), the asymptotic expressions

\[ \begin{aligned} w_1(t)&=e^{i\frac{\pi}{4}}(-t)^{-\frac14}e^{\,i\frac{2}{3}(-t)^{3/2}},\\ w_2(t)&=e^{-i\frac{\pi}{4}}(-t)^{-\frac14}e^{-\,i\frac{2}{3}(-t)^{3/2}}. \end{aligned} \tag{1.13} \]

Formula (1.10) also contains one of the functions \(u(t)\), \(v(t)\), defined by the equalities

\[ w_1(t)=u(t)+iv(t);\quad w_2(t)=u(t)-iv(t). \tag{1.14} \]

For real \(t\), both functions \(u(t)\), \(v(t)\) are real. For all values of \(t\) we have:

\[ w_1\left(te^{i\frac{2}{3}\pi}\right)=e^{i\frac{\pi}{3}}w_2(t);\quad w_1\left(te^{i\frac{4}{3}\pi}\right)=2e^{i\frac{\pi}{6}}v(t). \tag{1.15} \]

The contour \(C\) in the integral (1.09) encircles in the positive direction the first quadrant of the plane of the complex variable \(t\) (all poles of the integrand are located in the first quadrant). As the contour \(C\) we may take, for example, a broken line going from \(\infty e^{i\frac{2}{3}\pi}\) to \(0\) and from \(0\) to \(\infty\).

2. TRANSFORMATION OF THE ATTENUATION FACTOR

The attenuation factor \(V\) was investigated by us in our previous works \(^{1,2}\), first, in the illuminated region, where a reflected wave enters into force, corresponding to geometrical optics; second, in the shadow region, where the field amplitude decreases according to an exponential law; and, finally, in the transition region near the surface of the sphere (penumbra region). The region of the shadow cone, however, remained uninvestigated, and the conclusion-

of approximate formulas for this region constitutes the aim of the present work.

By the cone of shadow we shall mean the cone tangent to the sphere and having its vertex at the source. The equation of the cone of shadow may be written in the form

\[ \sqrt{b^{2}-a^{2}}+\sqrt{r^{2}-a^{2}}=\sqrt{r^{2}+b^{2}-2rb\cos\vartheta}, \tag{2.01} \]

or, after passing to the variables \(x, y_1, y_2\) and neglecting small quantities,

\[ \sqrt{y_1}+\sqrt{y_2}=x. \tag{2.02} \]

Thus we have to investigate the attenuation factor \(V\) for the case when the quantities \(x, y_1, y_2\) are very large, whereas the difference

\[ \xi=x-\sqrt{y_1}-\sqrt{y_2} \tag{2.03} \]

remains finite. Note that positive values of \(\xi\) correspond to the shadow region, and negative values to the illuminated region.

In the integral (1.09) for \(V\) we may understand by \(F\) either one of the two expressions (1.10) or (1.11), which are identically equal to each other. Let us split the contour \(C\) in the integral (1.09) into two parts: the part from \(\infty e^{i 2\pi/3}\) to \(0\) we denote by \(C_1\), and the part from \(0\) to \(\infty\) by \(C_2\). On the first part we shall use for \(F\) expression (1.11), and on the second part, expression (1.10). We may then write

\[ V=\Phi+\Psi, \tag{2.04} \]

where

\[ \Phi=\sqrt{\frac{x}{2}}\, e^{-i\pi/4} \left\{ \frac{i}{2}\int_{C_1} e^{ixt} w_1(t-y_2)w_2(t-y_1)\,dt + \int_{C_2} e^{ixt} w_1(t-y_2)v(t-y_1)\,dt \right\}, \tag{2.05} \]

\[ \Psi=-\sqrt{\frac{x}{\pi}}\, e^{-i\pi/4}\times \]

\[ \times \left\{ \frac{i}{2}\int_{C_1} e^{ixt} \frac{w_2'(t)-q w_2(t)}{w_1'(t)-q w_1(t)} w_1(t-y_1)w_1(t-y_2)\,dt + \right. \]

\[ \left. +\int_{C_2} e^{ixt} \frac{v'(t)-qv(t)}{w_1'(t)-q w_1(t)} w_1(t-y_1)w_1(t-y_2)\,dt \right\}. \tag{2.06} \]

The integrals entering into \(\Phi\) do not depend on the parameter \(q\), which enters only into \(\Psi\). Consequently, \(\Phi\) does not depend on the electrical properties of the body producing the diffraction; only \(\Psi\) depends on them. We shall see that \(\Phi\) corresponds to the Fresnel part of the diffraction, and \(\Psi\) to that background upon which the Fresnel diffraction pattern is superposed.

3. CALCULATION OF THE INTEGRAL \(\Phi\)

In expression (2.05) for \(\Phi\) we may replace integration over \(C_1\) by integration from \(-\infty\) to \(0\). Using the relation \(w_2=w_1-2iv\), we obtain:

\[ \Phi=\Phi_1+\Phi_2, \tag{3.01} \]

where

\[ \Phi_1=\frac{1}{2}\sqrt{\frac{x}{\pi}}\,e^{i\frac{\pi}{4}} \int_{-\infty}^{0} e^{ixt} w_1(t-y_2)w_1(t-y_1)\,dt, \tag{3.02} \]

\[ \Phi_2=\sqrt{\frac{x}{\pi}}\,e^{-i\frac{\pi}{4}} \int_{-\infty}^{+\infty} e^{ixt} w_1(t-y_2)v(t-y_1)\,dt. \tag{3.03} \]

Let us first calculate the integral \(\Phi_2\). For this we use the following integral representation for \(w_1(t-y_2)\):

\[ w_1(t-y_2)=\frac{1}{\sqrt{\pi}}\int_{\Gamma} e^{(t-y_2)z-\frac{1}{3}z^3}\,dz, \tag{3.04} \]

where the contour \(\Gamma\) consists of the sections from \(-i\infty\) to \(0\) and from \(0\) to \(\infty\). We note that on the contour \(\Gamma\) one has \(\operatorname{Re}(z)\geqslant 0\). After substituting (3.04) into (3.03), we can carry out the integration with respect to \(t\) by means of the formula

\[ \frac{1}{\sqrt{\pi}}\int_{-\infty}^{+\infty} e^{(z+ix)t}v(t-y_1)\,dt= \]

\[ =\exp\left\{y_1(z+ix)+\frac{1}{3}(z+ix)^3\right\}, \tag{3.05} \]

valid for \(\operatorname{Re}(z)\geqslant 0\). As a result we obtain:

\[ \Phi_2=\sqrt{\frac{x}{\pi}}\,e^{-i\frac{\pi}{4}}e^{-\frac{i}{3}x^3+ixy_1} \cdot \int_{\Gamma} e^{ixz^2-(x^2+y_2-y_1)z}\,dz. \tag{3.06} \]

The last integral is easily taken, and finally we obtain:

\[ \Phi_2=e^{i\omega(x)}, \tag{3.07} \]

where

\[ F^0(x)=-\frac{1}{12}x^3+\frac{1}{2}x(y_1+y_2)+\frac{(y_2-y_1)^2}{4x}. \tag{3.08} \]

As was shown in our paper\(^1\), the quantity \(\omega\) is the phase of the incident wave, approximately equal to

\[ \omega=k(R-s), \tag{3.09} \]

where \(R\) and \(s\) denote the same quantities as in Section 1. Thus the integral \(\Phi_2\) corresponds to the incident wave.

We proceed to the calculation of the integral \(\Phi_1\). Using the integral representation (3.04) for both factors \(w_1(t-y_2)\) and \(w_1(t-y_1)\) and carrying out the integration with respect to \(t\), we arrive at a double contour integral in which, after a change of variables, one integration can be performed. As a result one obtains

\[ \Phi_1=\frac{\sqrt{x}}{2\pi i}\int_C e^{i\omega(z)}\frac{dz}{\sqrt{z}(z-x)}, \tag{3.10} \]

where the contour \(C\) goes from positive imaginary infinity, crosses the real axis to the right of the point \(z=x\), and then goes along the ray \(\arg z=-\frac{\pi}{6}\).

The residue of the integral (3.10) at the point \(z=x\) is, according to (3.07), equal to the quantity \(\Phi_2\). Therefore, if we denote by \(C'\) a contour running analogously to \(C\), but crossing the real axis to the left of the point \(z=x\), we obtain:

\[ \Phi=\Phi_1+\Phi_2=\frac{\sqrt{x}}{2\pi i}\int_{C'} e^{i\omega(z)}\frac{dz}{\sqrt{z}(z-x)}. \tag{3.11} \]

With the aid of these formulas one can approximately express the function \(\Phi\) in terms of Fresnel integrals. For this purpose we shall apply the method of stationary phase, taking into account, however, that the fraction \(1/(z-x)\) will not be a slowly varying function.

Equating to zero the derivative of the phase \(\omega(z)\), we arrive at the equation

\[ z^4-2z^2(y_1+y_2)+(y_1-y_2)^2=0, \tag{3.12} \]

whose roots are

\[ z=\pm\sqrt{y_1}\pm\sqrt{y_2}. \tag{3.13} \]

Of these four roots we are interested only in the largest positive root

\[ z_0=\sqrt{y_1}+\sqrt{y_2}, \tag{3.14} \]

since it lies closest of all to the contour \(C\). Let us denote by \(C_0\) a contour analogous to \(C\) or \(C'\), but crossing the real ...

axis at the point \(z=z_0\). Applying the notation (2.03), we set

\[ x-z_0=x-\sqrt{y_1}-\sqrt{y_2}=\xi . \tag{3.15} \]

If \(\xi<0\), then the contour \(C_0\) is equivalent to \(C\), and the integral over it gives \(\Phi_1\). If, however, \(\xi>0\), then the contour \(C_0\) is equivalent to \(C'\), and the integral over it gives \(\Phi\).

Near \(z=z_0\) we have:

\[ \omega(z)=\omega_0-\mu^2(z-z_0)^2, \tag{3.16} \]

where

\[ \omega_0=\omega(z_0)=\frac{2}{3}y_1^{3/2}+\frac{2}{3}y_2^{3/2}, \tag{3.17} \]

\[ \mu^2=\frac{\sqrt{y_1y_2}}{\sqrt{y_1}+\sqrt{y_2}} . \tag{3.18} \]

For an approximate evaluation of the integral

\[ I=\frac{\sqrt{x}}{2\pi i}\int_{C_0} e^{i\omega(z)} \frac{dz}{\sqrt{z}(z-x)} \tag{3.19} \]

we replace the quantity \(\sqrt{z}\) by the constant value \(\sqrt{z_0}\), and the function \(\omega(z)\) by expression (3.16). Putting

\[ z=z_0+\rho e^{-i\frac{\pi}{4}}, \tag{3.20} \]

we can integrate with respect to \(\rho\) from \(-\infty\) to \(+\infty\). As a result one obtains

\[ I=\sqrt{\frac{x}{z_0}}\, e^{i\omega_0}\cdot \frac{1}{2\pi i}\int_{-\infty}^{+\infty} e^{-\mu^2\rho^2}\frac{d\rho}{\rho-\xi e^{i\frac{\pi}{4}}}. \tag{3.21} \]

The last integral is expressed in terms of Fresnel integrals; moreover, it has different analytic expressions for \(\xi>0\) and for \(\xi<0\), namely

\[ \frac{1}{2\pi i}\int_{-\infty}^{+\infty} e^{-\mu^2\rho^2}\frac{d\rho}{\rho-\xi e^{i\frac{\pi}{4}}} = \begin{cases} f(\mu\xi), & \text{for } \xi>0,\\ -f(-\mu\xi), & \text{for } \xi<0, \end{cases} \tag{3.22, 3.23} \]

where

\[ f(\alpha)=e^{-i\alpha^2-i\frac{\pi}{4}}\cdot \frac{1}{\sqrt{\pi}}\int_0^\infty e^{it^2}\,d\alpha . \tag{3.24} \]

It is easy to see that

\[ f(a)+f(-a)=e^{-ia^2}. \tag{3.25} \]

V. A. Fock

Introducing the usual Fresnel integrals

\[ C+iS=\sqrt{\frac{2}{\pi}}\int_0^a e^{i\alpha^2}\,d\alpha, \tag{3.26} \]

we can write

\[ f(\alpha)=\frac{1}{\sqrt{2}}\,e^{-i\alpha^2-i\frac{\pi}{4}} \left\{\left(\frac{1}{2}-C\right)+i\left(\frac{1}{2}-S\right)\right\}. \tag{3.27} \]

The asymptotic expression for \(f(\alpha)\), valid for large positive values of \(\alpha\), has the form:

\[ f(\alpha)=\frac{1}{2\sqrt{\pi}}\,e^{i\frac{\pi}{4}} \left(\frac{1}{\alpha}-\frac{i}{2\alpha^3}+\cdots\right). \tag{3.28} \]

Expressing the integral \(I\) through \(f(\alpha)\) and recalling that this integral represents, for \(\xi>0\), the function \(\Phi\), and for \(\xi<0\) the function \(\Phi_1=\Phi-\Phi_2\), where \(\Phi_2\) is defined by (3.07), we finally obtain:

\[ \Phi=\frac{\sqrt{x}}{\sqrt[4]{y_1y_2}}\,e^{i\omega_0}\,\mu f(\mu\xi) \qquad (\text{for } \xi>0), \tag{3.29} \]

\[ \Phi=e^{i\omega(x)}-\frac{\sqrt{x}}{\sqrt[4]{y_1y_2}}\,e^{i\omega_0}\,\mu f(-\mu\xi) \qquad (\text{for } \xi<0). \tag{3.30} \]

These expressions are valid under the condition that both numbers \(\sqrt{y_1}\) and \(\sqrt{y_2}\) are very large (the quantity \(\mu^2\) will be of the order of the smaller of these numbers). As for the quantity \(\xi\), it may be either finite or small, and the product \(\mu\xi\) may be any number (large, finite, or small). If \(\xi\) is very small (and it may be of either sign), then the two expressions for \(\Phi\) practically coincide. This is seen from the approximate equalities

\[ \frac{\mu^2x}{\sqrt{y_1y_2}}=1+\frac{\xi}{\sqrt{y_1}+\sqrt{y_2}}\sim 1, \tag{3.31} \]

\[ \omega(x)\sim \omega_0-\mu^2\xi^2 \tag{3.32} \]

in conjunction with formula (3.25). For \(\xi=0\) the coincidence of both expressions for \(\Phi\) will be exact.

4. EVALUATION OF THE INTEGRAL \(\Psi\)

Let us now turn to the derivation of approximate formulas for the integral \(\Psi\). We are interested in the value of the integral for the same case for which we evaluated the integral \(\Phi\), namely, for the case when the quantities \(\sqrt{y_1}\), \(\sqrt{y_2}\) (and, consequently, \(\mu^2\)) are very large, while the quantity \(\xi=x-\sqrt{y_1}-\sqrt{y_2}\) is finite. Under these conditions the principal part of the integration will be that where the variable \(t\)

finite. But for finite \(t\) and large \(y_1\) and \(y_2\), the product of the function \(w_1\) by the exponential function, standing under the integral in (2.06), will be equal to

\[ e^{ixt} w_1(t-y_1) w_1(t-y_2) = -\frac{i}{\sqrt[4]{y_1 y_2}}\, e^{i\omega_0}\cdot e^{i\xi t} \left(1+\frac{i t^2}{4\mu^2}+O\left(\frac{1}{\mu^4}\right)\right), \tag{4.01} \]

where, for brevity, we have used the notation (3.17).

Substituting this expression into the integral \(\Psi\), we obtain:

\[ \Psi = -\frac{\sqrt{x}}{\sqrt[4]{y_1 y_2}}\, e^{i\omega_0} \left\{ g(\xi)-\frac{i}{4\mu^2}g''(\xi)+O\left(\frac{1}{\mu^4}\right) \right\}, \tag{4.02} \]

where

\[ g(\xi)=\frac{1}{\sqrt{\pi}} e^{i\frac{\pi}{4}} \left\{ \frac{i}{2} \int_{\infty e^{i\frac{2}{3}\pi}}^{0} e^{i\xi t}\, \frac{w_2'(t)-q w_2(t)}{w_1'(t)-q w_1(t)}\,dt + \int_{0}^{\infty} e^{i\xi t}\, \frac{v'(t)-qv(t)}{w_1'(t)-q w_1(t)}\,dt \right\}. \tag{4.03} \]

Using the properties of the Airy functions (1.15), it is easy to verify that if

\[ t=t'e^{i\frac{2\pi}{3}}, \tag{4.04} \]

then

\[ \frac{i}{2}\, \frac{w_2'(t)-qw_2(t)}{w_1'(t)-qw_1(t)} = \frac{v'(t')-q e^{i\frac{2}{3}\pi}v(t')} {w_2'(t')-q e^{i\frac{2}{3}\pi}w_2(t')}. \tag{4.05} \]

The substitution (4.04) reduces the first integral in (4.03) to an integral along the real positive axis. Omitting the prime on \(t\), we obtain:

\[ g(\xi) = e^{-i\frac{\pi}{12}}\cdot\frac{1}{\sqrt{\pi}} \int_{0}^{\infty} e^{-\frac{\xi t}{2}(\sqrt{3}+i)} \cdot \frac{v'(t)-q e^{i\frac{2}{3}\pi}v(t)} {w_2'(t)-q e^{i\frac{2}{3}\pi}w_2(t)}\,dt + \]

\[ \qquad + e^{i\frac{\pi}{4}}\frac{1}{\sqrt{\pi}} \int_{0}^{\infty} e^{i\xi t}\cdot \frac{v'(t)-qv(t)} {w_1'(t)-q w_1(t)}\,dt. \tag{4.06} \]

As \(t\) increases, the function \(v(t)\) in the numerator rapidly decreases, whereas the functions \(w_1(t)\) and \(w_2(t)\) in the denominator increase just as rapidly. Therefore both integrals converge very rapidly and can be evaluated by quadratures. The function \(g(\xi)\) admits an expansion in a Taylor series in powers of \(\xi\); the coefficients of this series can also be computed by quadratures. For large positive values of \(\xi\), the function \(g(\xi)\) has the asymptotic expression

\[ g(\xi)=\frac{e^{i\frac{\pi}{4}}}{2\sqrt{\pi}}\cdot\frac{1}{\xi}, \tag{4.07} \]

which reduces to a single term, independent in this case of \(q\). The remainder will be of order \(e^{i\xi t_1}\), where \(t_1\) is the first root of the equation

\[ w_1'(t)-q w_1(t)=0. \tag{4.08} \]

For large negative \(\xi\), the asymptotic expression for \(g(\xi)\) has the form:

\[ g(\xi)=\frac{e^{i\frac{\pi}{4}}}{2\sqrt{\pi}}\cdot\frac{1}{\xi} +\frac{\sqrt{-\xi}}{2}\cdot\frac{q+i\frac{\xi}{2}}{q-i\frac{\xi}{2}}e^{-\frac{i}{12}\xi^2}. \tag{4.09} \]

When substituting this expression into (4.02), it must be borne in mind that this formula for \(\Psi\) is applicable in the case when the correction term containing \(\mu^2\) in the denominator is small in comparison with the principal term. For both expressions (4.02) and (4.09) to be applicable, the condition

\[ 1\ll \xi^2\ll \mu \qquad (\xi<0) \tag{4.10} \]

must be satisfied.

5. THE ATTENUATION FACTOR IN THE REGION OF THE SHADOW CONE

In the preceding sections we found approximate expressions for the integrals \(\Phi\) and \(\Psi\), the sum of which gives the attenuation factor \(V(x,y_1,y_2,q)\). Forming the sum, we obtain for \(\xi\gg0\)

\[ V=\frac{\sqrt{x}}{\sqrt[4]{y_1y_2}}e^{i\omega_0} \left\{\mu f(\mu\xi)-g(\xi)+\frac{i}{4\mu^2}g''(\xi)\right\} \tag{5.01} \]

and for \(\xi\ll0\)

\[ V=e^{i\omega(x)}-\frac{\sqrt{x}}{\sqrt[4]{y_1y_2}}e^{i\omega_0} \left\{\mu f(-\mu\xi)+g(\xi)-\frac{i}{4\mu^2}g''(\xi)\right\}. \tag{5.02} \]

These expressions are valid under the condition that the parameter \(\mu\), determined from the equality

\[ \mu^2=\frac{\sqrt{y_1y_2}}{\sqrt{y_1}+\sqrt{y_2}}, \tag{5.03} \]

the parameter \(\mu\) is very large, whereas the quantity

\[ \xi=x-\sqrt{y_1}-\sqrt{y_2} \tag{5.04} \]

is finite or small.

Let us recall the geometrical meaning of these quantities. According to formulas (1.03)—(1.05) we have:

\[ \mu^2=\sqrt[6]{\frac{2k^2}{a}}\cdot \frac{\sqrt{h_1 h_2}}{\sqrt{h_1}+\sqrt{h_2}}, \tag{5.05} \]

\[ \xi=\sqrt[3]{\frac{k}{2a^2}}\left(s-\sqrt{2ah_1}-\sqrt{2ah_2}\right). \tag{5.06} \]

Thus, large values of \(\mu\) correspond to small wavelengths and to relatively large distances from the surface of the body (the latter must still be small in comparison with the radii of its curvature). The quantity \(\xi\) is proportional to the distance from the geometrical boundary of the shadow (the shadow cone), measured along (more precisely, parallel to) the surface of the body. For \(\xi<0\) the quantity \(\mu^2\xi^2\) is approximately equal to the phase difference between the reflected and the incident waves. The value \(\xi=0\) corresponds to the boundary of the shadow; positive values of \(\xi\) correspond to the shadow region, and negative values to the illuminated region.

Our formulas give the transition from light to shadow at relatively large distances from the surface of the body. Since the functions \(f\) and \(g\), and their derivatives with respect to their arguments, will be of order unity for finite values of the arguments, for large values of \(\mu\) the leading term in (5.01) will be the term \(\mu f(\mu\xi)\). This term is proportional to the Fresnel integral. It represents a rapidly varying function of \(\xi\), since the argument in the Fresnel integral is \(\mu\xi\), where \(\mu\) is a large number. Thus the leading term in the expression for \(V\) gives Fresnel diffraction. But upon this diffraction pattern there is superposed a background represented by the function \(g(\xi)\), which, in comparison with the leading term, varies slowly. This background depends on the material of the diffracting body (since \(g(\xi)\) depends on \(q\)), whereas the Fresnel term does not depend on it.

The formulas obtained here for the attenuation factor must, when one moves in either direction away from the shadow cone, pass into the formulas previously derived by us for the shadow and illuminated regions. Let us verify this. In the shadow region we must obtain an exponential decrease of the amplitude, and in the illuminated region, a reflection formula. Since in formula (5.01), and in the asymptotic expression (4.07) for \(g(\xi)\), the terms that decrease exponentially for large positive \(\xi\) are neglected on account of their smallness, we must obtain zero in the shadow region in our approximation. Indeed, from the asymptotic expression (3.28) for the Fresnel function \(f(c)\)

it follows:

\[ \mu f(\mu \xi)=-\frac{1}{2\sqrt{\pi}}\,e^{i\frac{\pi}{4}}\cdot \left(\frac{1}{\xi}-\frac{i}{2\mu^2\xi^3}\right). \tag{5.07} \]

On the other hand, formula (4.07) gives:

\[ g(\xi)-\frac{i}{4\mu^2}g''(\xi) =-\frac{1}{2\sqrt{\pi}}\,e^{i\frac{\pi}{4}}\cdot \left(\frac{1}{\xi}-\frac{i}{2\mu^2\xi^3}\right), \tag{5.08} \]

i.e., the same expression. Thus, for large positive \(\xi\), expression (5.01) for \(V\) does indeed vanish in our approximation.

Let us now consider large negative values of \(\xi\). In formula (5.02), the first term of the asymptotic expression (4.09) for \(g(\xi)\) cancels with \(\mu f(-\mu\xi)\), while the second term (containing the indicator function) gives:

\[ V=e^{i\omega(x)}-\frac{\sqrt{x}}{\sqrt[4]{y_1y_2}}\,e^{i\omega_0}\cdot \frac{\sqrt{-\xi}}{2}\cdot \frac{q+i\frac{\xi}{2}}{q-i\frac{\xi}{2}}\, e^{-\frac{i}{12}\xi^3}. \tag{5.09} \]

On the other hand, as was shown in our work\(^1\), in the illuminated region the reflection formula holds:

\[ V=e^{i\omega}\cdot\left(1-\frac{q-ip}{q+ip}\cdot \sqrt{\frac{p}{p+p_1}}\,e^{2ipp_1}\right) \tag{5.10} \]

(formula (4.31) of the cited work). Here \(\omega=\omega(x)\), and the quantity \(p\) (proportional to the cosine of the angle of incidence) is determined from the equation

\[ \sqrt{y_1+p^2}+\sqrt{y_2+p^2}=2p+x, \tag{5.11} \]

while the quantity \(p_1\) is equal to

\[ p_1=2p+x-\frac{1}{x}(y_1+y_2). \tag{5.12} \]

In the approximation in which formula (5.09) is valid,

\[ p=-\frac{\xi}{2}+\frac{\xi^2}{16\mu^2}\sim-\frac{\xi}{2}, \tag{5.13} \]

\[ p_1=2\mu^2+\xi-\frac{2\mu^2\xi}{x}\sim 2\mu^2. \tag{5.14} \]

Using these approximate equalities, it is not difficult to verify that formula (5.09) represents the approximate form of the reflection formula (5.10).

Thus, formulas (5.01) and (5.02), derived for the region close to the shadow cone, join up with formulas valid in the regions adjacent on both sides to the shadow cone and derived in our previous works.

In conclusion, let us make several remarks concerning the formulas derived here.

As the initial expression for \(V\), as well as the approximate formulas, permit, with a corresponding change in the expression for the phase of the incident wave, passage to the case of a plane wave. This passage reduces to our increasing \(x\) and \(\sqrt{y_2}\) to infinity, while leaving their difference finite. But, as was shown in our papers² and ³, in the case of a plane wave our initial formulas are valid not only for a sphere, but also for a body of arbitrary shape. Therefore the approximate formulas derived here, containing Fresnel integrals, may be regarded as proved also for a body of arbitrary shape. It also appears very probable that the diffraction pattern obtained here (Fresnel diffraction, upon which a background is superposed) occurs, at least qualitatively, also at large distances from the body. In this case one should expect that the background becomes weaker as the distance from the body increases.

References Cited

1. V. A. Fock, Field of a vertical and horizontal dipole raised above the surface of the earth, ZhETF 19, 916 (1949).
2. V. A. Fock, Field of a plane wave near the surface of a conducting body, Izvestiya AN SSSR, ser. fiz. 10, 171 (1946).
3. V. A. Fock, The laws of Fresnel reflection and the laws of diffraction, UFN 36, 308 (1948).