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FRESNEL LAWS OF REFLECTION
AND LAWS OF DIFFRACTION
V. A. Fok
In 1821 the French scientist Fresnel established formulas determining the intensities and directions of the oscillations in the reflected and refracted ray of light incident on the plane surface of a transparent body.
Fresnel obtained his formulas on the basis of the elastic theory of light, under the assumption of transverse oscillations of an elastic medium (the ether), and in doing so he had to introduce special hypotheses concerning the elasticity and density of the ether in media differing from one another in refractive index. This derivation does not accord with modern views on the nature of light and at present is of historical interest only. The formulas themselves, however, were brilliantly confirmed by experiment and subsequently served as a touchstone for testing every new theory of light.
In 1865 there appeared Maxwell’s electromagnetic theory of light, which withstood this test and, moreover, gave an explanation of an unusually wide range of phenomena, including those discovered many years later, such as radio waves (Hertz, Popov), light pressure (Lebedev), and many others.
Fresnel’s laws of reflection follow, without any additional hypotheses, from Maxwell’s equations and the corresponding boundary conditions; moreover, it turns out that by the transverse oscillations considered by Fresnel one must understand oscillations of the electric vector.
Fresnel’s laws are applicable not only to light, but also to electromagnetic oscillations of any frequency, including radio waves. On the other hand, Fresnel’s laws are easily generalized to the case where waves are incident on the plane surface of an absorbing body. Fresnel’s formulas retain their form here, with the sole difference that the refractive index \(n\) must be replaced by a complex quantity, namely, the square root of the complex dielectric constant of the medium.
Fresnel’s formulas directly make it possible to express the amplitudes of the electromagnetic field of the reflected wave in terms of the amplitudes of the field of the incident wave, where by both amplitudes one means their values on the reflecting surface. If
if a plane wave is incident on a surface and if the reflecting surface itself is plane, then the amplitudes of the field of the reflected wave at some distance from the surface will be the same as at the surface itself; only the phase will depend on the distance from the surface. If, however, the reflecting surface is convex, then the incident parallel beam of rays becomes divergent after reflection. In that case, when calculating the amplitude of the reflected wave at a given distance from the point where reflection occurred, it is necessary to introduce into the amplitude a correction factor that takes account of the expansion of the beam after reflection. This factor can be found from purely geometrical considerations.
The laws of reflection of an electromagnetic wave found a very simple and convenient approximate formulation in the Fresnel formulas. The situation was far less satisfactory with the approximate formulation of the laws of diffraction, i.e., the bending of waves around obstacles and their penetration into the region of geometrical shadow. All approximate methods known until recently concerned the case of diffraction of waves by obstacles with sharp edges, for example by opaque screens with apertures. These methods are, in essence, refinements of Huygens’ principle. The main step in this direction was taken by Fresnel himself. According to Huygens’ principle in Fresnel’s formulation, that part of a light wave which is covered by a screen does not act at all, while the uncovered regions act as if there were no screen at all. A further refinement was made in 1882 by Kirchhoff, who proposed his formula for the amplitude of the light wave behind a screen. Kirchhoff’s formula is a very flexible and convenient tool for the approximate solution of diffraction problems involving a screen with sharp edges, but it does not take into account the influence of the material of the screen and, in general, does not take account of the boundary conditions for the field that follow from Maxwell’s theory.
The next essential step in solving the problem of diffraction by a screen with sharp edges is connected with the finding of rigorous solutions of Maxwell’s equations for certain special cases (half-plane, wedge). Here one should mention the works of Sommerfeld, as well as the works of the Soviet mathematicians S. L. Sobolev and V. I. Smirnov, who approached the same problem from a new point of view (nonstationary processes). Extremely interesting problems concerning plane and cylindrical waveguides with open ends (where the diffracted wave can bend backward) have recently been solved by the young Soviet scientist L. A. Weinstein.
In contrast to the problem of diffraction by bodies with sharp edges (screens and diaphragms), for the solution of the problem of diffraction by bodies with continuously varying curvature no reasonably general approximate methods or approximate formulas (similar to Kirchhoff’s formula) had been proposed until the very latest time. To find the field obtained as a result of the diffraction of an incident wave, it was proposed, for each separate case,
solve Maxwell’s equations with boundary conditions, which constitutes a very difficult mathematical problem.
Fresnel’s reflection formulas constitute an integral law in the sense that their application does not require solving differential equations, since these formulas give explicit expressions for the amplitudes of the reflected wave. For the phenomenon of diffraction by a body of arbitrary shape, not only was the form of the corresponding integral law unknown, but even the fact of the existence of such a law had not been established; in other words, the very possibility had not been established of writing, under some sufficiently general assumptions about the electrical properties of the material of the body and about the shape of its surface, explicit expressions for the amplitudes of the field of the wave bending around the body.
This gap was to a certain extent filled in our works on the diffraction of a plane wave by the surface of a convex conducting body of arbitrary shape.
The assumption that the material of the body is a good conductor is essential, because it makes it possible to use the simplified boundary conditions for the field established by M. A. Leontovich.
Considering the field near the surface of the body (at distances small in comparison with the radii of curvature of the surface), we established that in the penumbra region this field has a local character. This means that, for a given wavelength of the incident wave, its amplitude and polarization, the field in the penumbra region depends only on the shape and properties of the body near the given point, and is expressed through certain universal functions which can be tabulated once and for all. Thus it becomes possible to formulate a certain general law of diffraction.
Our formulas for the field may be regarded as a generalization of Fresnel’s formulas—a generalization which includes both the law of reflection and the law of diffraction.
Let us mentally move along the surface of the body from its illuminated side into the shadow. On the illuminated side of the body one can distinguish the incident and reflected waves, the latter being well described by Fresnel’s formulas. Closer to the geometrical boundary of the shadow, in the region of grazing incidence of the ray, the two waves are no longer separable from one another, so that it makes sense to consider only the resultant field. Here our formulas come into force, whereas Fresnel’s formulas become inapplicable. Beyond the boundary of the geometrical shadow we no longer have a wave with a more or less constant amplitude, but have a damped wave, i.e. a wave whose amplitude decreases according to an exponential law as the distance from the geometrical boundary of the shadow increases. Here the phenomenon of diffraction in the proper sense takes place, and the law of diffraction is conveyed by our formulas.
From what has been said it is clear that there exists a region (namely, the region of grazing incidence of the ray) where both our
diffraction formulas, as well as Fresnel formulas. It is obvious that in this region the one formulas must pass into the other.
In what follows we shall give the Fresnel formulas for the electromagnetic field and indicate their generalization, which makes it possible to take into account the spreading of a beam after reflection from a convex body. Further, we shall write out the diffraction formulas obtained by us, consider their limiting cases, and trace how they pass into the Fresnel formulas in the region of grazing incidence of the ray.
1. FRESNEL LAWS OF REFLECTION
Let us denote by \(\mathbf E^0(E_x^0,E_y^0,E_z^0)\) and by \(\mathbf H^0(H_x^0,H_y^0,H_z^0)\) the amplitudes of the electric and magnetic vectors of the incident wave at a given point of the surface of the body. The corresponding quantities for the reflected wave we shall denote by \(\mathbf E^*(E_x^*,E_y^*,E_z^*)\) and \(\mathbf H^*(H_x^*,H_y^*,H_z^*)\). Further, let \(\mathbf a(a_x,a_y,a_z)\) be the unit vector in the direction of the incident ray, \(\mathbf a^*(a_x^*,a_y^*,a_z^*)\) the unit vector in the direction of the reflected ray, and \(\mathbf n(n_x,n_y,n_z)\) the unit vector of the normal to the surface of the body at the point of incidence. According to the law of reflection, the quantities \(\mathbf a^*\), \(\mathbf a\), and \(\mathbf n\) are connected by the relation
\[ \mathbf a^*=\mathbf a-2\mathbf n(\mathbf a\cdot\mathbf n), \tag{1,01} \]
where
\[ \mathbf a^*\cdot\mathbf n=-\,\mathbf a\cdot\mathbf n=\cos\vartheta, \tag{1,02} \]
where \(\vartheta\) is the angle of incidence. The quantities \(\mathbf a\) and \(\mathbf a^*\) are proportional to the gradient of the phase of the incident and reflected wave. Considering the amplitude to be a quantity varying slowly in comparison with the phase, we obtain from Maxwell’s equations for empty space
\[ [\mathbf a\times \mathbf E^0]=\mathbf H^0;\qquad \mathbf a\cdot\mathbf E^0=0, \tag{1,03} \]
whence
\[ [\mathbf a\times \mathbf H^0]=-\mathbf E^0;\qquad \mathbf a\cdot\mathbf H^0=0, \tag{1,04} \]
and, analogously, for the reflected wave
\[ [\mathbf a^*\times \mathbf E^*]=\mathbf H^*;\qquad \mathbf a^*\cdot\mathbf E^*=0, \tag{1,05} \]
\[ [\mathbf a^*\times \mathbf H^*]=-\mathbf E^*;\qquad \mathbf a^*\cdot\mathbf H^*=0. \tag{1,06} \]
Let us denote by \(\mu\) the magnetic permeability, and by
\[ \eta=\varepsilon+i\,\frac{4\pi\sigma}{\omega} \tag{1,07} \]
the complex dielectric constant of the substance of the reflecting body, and introduce the Fresnel coefficients
\[ N=\frac{\eta\cos\vartheta-\sqrt{\mu\eta-\sin^2\vartheta}} {\eta\cos\vartheta+\sqrt{\mu\eta-\sin^2\vartheta}}, \tag{1,08} \]
\[ M=\frac{\mu\cos\vartheta-\sqrt{\mu\eta-\sin^2\vartheta}} {\mu\cos\vartheta+\sqrt{\mu\eta-\sin^2\vartheta}}. \tag{1,09} \]
Then the Fresnel formulas, establishing the relation between the amplitudes of the incident and reflected waves, may be written in the form
\[ (\mathbf n\cdot \mathbf E^*)=N(\mathbf n\cdot \mathbf E^0), \tag{1,10} \]
\[ (\mathbf n\cdot \mathbf H^*)=M(\mathbf n\cdot \mathbf H^0). \tag{1,11} \]
The amplitudes of the transmitted wave (penetrating into the material of the body) do not interest us, and we shall not write out the corresponding formulas. Equations (1,05), (1,10), and (1,11) can be solved with respect to the vectors \(\mathbf E^*\) and \(\mathbf H^*\). Introducing the notation
\[ \mathbf n\cdot \mathbf E^0=E_n^0,\qquad \mathbf n\cdot \mathbf H^0=H_n^0 \tag{1,12} \]
and expressing \(\mathbf a^*\) in terms of \(\mathbf a\) according to (1,01), we shall have
\[ \sin^2\vartheta\, \mathbf E^* = -NE_n^0(\mathbf n\cos 2\vartheta+\mathbf a\cos\vartheta) +MH_n^0[\mathbf n\times \mathbf a], \tag{1,13} \]
\[ \sin^2\vartheta\, \mathbf H^* = -MH_n^0(\mathbf n\cos 2\vartheta+\mathbf a\cos\vartheta) -NE_n^0[\mathbf n\times \mathbf a]. \tag{1,14} \]
Such are the values of the amplitudes of the reflected wave on the surface of the body which follow from the Fresnel formulas.
From the preceding formulas one can also derive relations for the total field. Denoting by \(\mathbf E,\mathbf H\) the total field on the surface of the body and by \(E_n\) and \(H_n\) its normal components, and putting
\[ \chi=\sqrt{\,1-\frac{\sin^2\vartheta}{\eta\mu}\,}, \tag{1,15} \]
we shall have
\[ \sin^2\vartheta\,(\mathbf E-\mathbf n E_n) = \chi\sqrt{\frac{\mu}{\eta}}\,E_n\{\mathbf a-\mathbf n(\mathbf a\cdot \mathbf n)\} +H_n[\mathbf n\times \mathbf a], \tag{1,16} \]
\[ \sin^2\vartheta\,[\mathbf n\times \mathbf H] = E_n\{\mathbf a-\mathbf n(\mathbf a\cdot \mathbf n)\} +\chi\sqrt{\frac{\eta}{\mu}}\,H_n[\mathbf n\times \mathbf a]. \tag{1,17} \]
If \(|\eta\mu|\gg 1\), then approximately \(\chi=1\), and the right-hand sides of (1,16) and (1,17) are proportional to one another. In this case
\[ \mathbf E-\mathbf nE_n=\sqrt{\frac{\mu}{\eta}}\,[\mathbf n\times \mathbf H]. \tag{1,18} \]
The last relation no longer contains the vector \(\mathbf a\), i.e., it does not depend on the direction of the incident wave. As M. A. Leontovich showed, it holds not only in the illuminated region, where the Fresnel formulas are applicable, but also on the whole surface of the body.
From formulas (1,16) and (1,17) one can also derive the relations:
\[ (\mathbf a\cdot \mathbf E)= \left(-\cos\vartheta+\chi\sqrt{\frac{\mu}{\eta}}\right)E_n, \tag{1,19} \]
\[ (\mathbf a\cdot \mathbf H)= \left(-\cos\vartheta+\chi\sqrt{\frac{\eta}{\mu}}\right)H_n. \tag{1,20} \]
If the incident wave is plane, so that the vector \(\mathbf a\) has a definite value, then the latter relations may be used instead of Leontovich’s conditions (1,18). This is convenient to do when grazing incidence of a ray is considered; moreover, in expression (1,15) for \(z\) one may put \(\sin^2 \vartheta = 1\).
2. CROSS SECTION OF A BEAM OF REFLECTED RAYS
To find the amplitude of the reflected wave at some distance from the surface of the body, it is necessary to have formulas for the cross section of a beam resting on an element \(dS\) of the surface of the body and traversing, after reflection, a prescribed path \(s\). These formulas can be derived from the known formulas of differential geometry.
Let the equation of the reflecting surface be
\[ x = x_0(u, v);\qquad y = y_0(u, v);\qquad z = z_0(u, v), \tag{2,01} \]
where \(u, v\) are Gaussian curvilinear coordinates. We shall write the square of the line element on the surface in the form
\[ dl^2 = g_{uu}du^2 + 2g_{uv}du\,dv + g_{vv}dv^2 = \sum_{u,v} g_{uv}du\,dv, \tag{2,02} \]
where the sum \(\sum_{u,v}\) is an abbreviated notation for the middle term of this equality.
We shall use notations for the covariant and contravariant components of vectors and tensors, raising and lowering indices with the aid of the “metric” tensor entering into (2,02). We shall write the surface element in the form
\[ dS = \sqrt{g}\,du\,dv. \tag{2,03} \]
Let us write out the formulas for the components of the vector normal to the surface and their derivatives with respect to \(u, v\). We have
\[ \sqrt{g}\,n_x = \frac{\partial y_0}{\partial u}\frac{\partial z_0}{\partial v} - \frac{\partial y_0}{\partial v}\frac{\partial z_0}{\partial u} \quad \text{etc.} \tag{2,04} \]
\[ \frac{\partial n_x}{\partial u} = -\sum G_u^v \frac{\partial x_0}{\partial v} \quad \text{etc.} \tag{2,05} \]
The last formula may serve as the definition of the quantities \(G_u^v\)—the mixed components of the second quadratic form of the surface. If \(R_1\) and \(R_2\) are the principal radii of curvature of the normal section of the surface, then we shall have
\[ K = \frac{1}{R_1R_2} = G_u^uG_v^v - G_v^uG_u^v, \tag{2,06} \]
\[ \frac{1}{R_1} + \frac{1}{R_2} = -G = -G_u^u - G_v^v. \tag{2,07} \]
The quantity \(K\) is the Gaussian curvature of the surface. We shall need a formula for the radius of curvature \(R_0\) of the normal section of the surface by the plane of incidence of the ray. It can be shown that if \(k\psi\) is the phase of the incident wave, with
\[ (\operatorname{grad}\psi)^2=1, \tag{2.08} \]
then
\[ \sum_{u,v} g^{uv}\,\frac{\partial \psi_0}{\partial u}\frac{\partial \psi_0}{\partial v} =\sin^2\vartheta, \tag{2.09} \]
where \(\vartheta\) is the angle of incidence, and the derivatives are taken at the value \(\psi=\psi_0\) of the phase on the surface of the body. The quantity \(R_0\) is then determined from the equation
\[ \sum_{u,v} G^{uv}\,\frac{\partial \psi_0}{\partial u}\frac{\partial \psi_0}{\partial v} =-\,\frac{\sin^2\vartheta}{R_0}. \tag{2.10} \]
Let us apply the formulas written out here to the computation of the normal section of a beam of rays reflected from the surface element \(dS\).
Consider the equations:
\[ \begin{aligned} x&=x_0+s a_x^*,\\ y&=y_0+s a_y^*,\\ z&=z_0+s a_z^*, \end{aligned} \tag{2.11} \]
in which \(s\) is some prescribed quantity, while \(x_0,y_0,z_0,a_x^*,a_y^*,a_z^*\) are functions of \(u,v\), determined from the equation of the surface (2.01) and from the relations
\[ \mathbf{a}^*=\mathbf{a}-2\mathbf{n}(\mathbf{a}\cdot\mathbf{n}), \tag{2.12} \]
where \(\mathbf{n}\) is the normal vector at the point \(x_0,y_0,z_0\).
The quantity \(s\) is, evidently, the path traversed by the ray after reflection. For constant \(s\), equations (2.11) represent the equations of a certain surface, in a known sense parallel to the reflecting surface of the body. If we vary \(u,v\) within the limits \((u,u+du)\), \((v,v+dv)\), we obtain a certain element of the surface (2.11). This element may be regarded as a section by the surface of the beam of reflected rays resting on the element \(dS=\sqrt{g}\,du\,dv\). In order to obtain the normal section of the beam, we must project this element onto the plane perpendicular to the reflected ray. Denoting the area of the normal section by \(D(s)dS\), we shall have
\[ D(s)dS= \begin{vmatrix} a_x^* & a_y^* & a_z^*\\ \dfrac{\partial x}{\partial u} & \dfrac{\partial y}{\partial u} & \dfrac{\partial z}{\partial u}\\ \dfrac{\partial x}{\partial v} & \dfrac{\partial y}{\partial v} & \dfrac{\partial z}{\partial v} \end{vmatrix} \,du\,dv, \tag{2.13} \]
whence
\[ D(s)=\frac{1}{\sqrt{g}} \begin{vmatrix} a_r^{*} & a_y^{*} & a_z^{*}\\ \dfrac{\partial x}{\partial u} & \dfrac{\partial y}{\partial u} & \dfrac{\partial z}{\partial u}\\ \dfrac{\partial x}{\partial v} & \dfrac{\partial y}{\partial v} & \dfrac{\partial z}{\partial v} \end{vmatrix}. \tag{2,14} \]
We shall compute this determinant under the assumption that the incident wave is plane and that, consequently, the vector \(\mathbf a\) does not depend on \(u, v\). After rather complicated calculations, which we omit here, the following result is obtained:
\[ D(s)=\cos\vartheta+2s\left( -G+G\sum_{u,v} g^{uv}\frac{\partial\psi_0}{\partial u}\frac{\partial\psi_0}{\partial v} -\sum_{u,v}G^{uv}\frac{\partial\psi_0}{\partial u}\frac{\partial\psi_0}{\partial v} \right) +4Ks^2\cos\vartheta . \tag{2,15} \]
Using the expressions given above (2,06)—(2,10), we may write:
\[ D(s)=\cos\vartheta+2s\left[ \left(\frac{1}{R_1}+\frac{1}{R_2}\right)\cos^2\vartheta +\frac{\sin^2\vartheta}{R_0} \right] +\frac{4s^2}{R_1R_2}\cos\vartheta, \tag{2,16} \]
where the values \(R_0, R_1, R_2\) are taken at the point where the reflection occurred. The quantity \(D(s)/D(0)\) evidently gives the expansion of the beam, i.e. the ratio of its cross-section at a distance \(s\) from the surface (measured along the ray) to the cross-section at the surface itself.
3. THE ELECTROMAGNETIC FIELD OF THE REFLECTED WAVE
Let the field of the incident plane wave be equal to
\[ \mathbf E^0 e^{i\varphi},\quad \mathbf H^0 e^{i\varphi}, \tag{3,01} \]
where \(\mathbf E^0\) and \(\mathbf H^0\) are constant amplitudes and
\[ \varphi=k\phi=k(xa_x+ya_y+za_z) \tag{3,02} \]
is the phase of the wave at the given point of space. Introducing the value
\[ \varphi_0=k\phi_0=k(x_0a_x+y_0a_y+z_0a_z) \tag{3,03} \]
of the phase \(\varphi\) on the surface of the body, we shall have, for the field of the incident wave on the surface of the body, the expressions
\[ \mathbf E^0 e^{ik\phi_0},\quad \mathbf H^0 e^{ik\phi_0}. \tag{3,04} \]
The field of the reflected wave on the surface of the body will be equal to
\[ \mathbf E^{*}e^{ik\phi_0},\quad \mathbf H^{*}e^{ik\phi_0}, \tag{3,05} \]
where \(\mathbf E^*\) and \(\mathbf H^*\) are related to \(\mathbf E^0\) and \(\mathbf H^0\) by the Fresnel formulas (1.13) and (1.14). (Concerning the notation, we note that in formulas (1.13) and (1.14) we regarded the phase factor \(e^{ik\psi}\) as included in \(\mathbf E^0, \mathbf H^0\) and in \(\mathbf E^*, \mathbf H^*\); but since this factor is the same in both sides of equalities (1.13) and (1.14), it is immaterial whether in these equalities we understand by \(\mathbf E^0, \mathbf H^0\) and \(\mathbf E^*, \mathbf H^*\) the complete expressions (3.04) and (3.05) or their amplitudes.)
In the notation of this paragraph \(\mathbf E^0\) and \(\mathbf H^0\) are constants, while \(\mathbf E^*\) and \(\mathbf H^*\) are slowly varying functions of the coordinates of the point on the surface. Denote by \(F\) one of the components of the field of the reflected wave. The value of \(F\) on the surface will be equal to
\[ F=f(u,v)e^{ik\psi_0(u,v)}, \tag{3.06} \]
where \(f(u,v)\) is a slowly varying function, and \(k\) is a large parameter. To find the value of \(F\) at some distance \(s\) from the surface, we must know the solution of the wave equation
\[ \Delta F+k^2F=0, \tag{3.07} \]
which satisfies the radiation condition and the boundary condition (3.06) on the surface. Using the fact that \(k\) is a large parameter, one can indicate an approximate form of such a solution explicitly.
Indeed, consider the expression
\[ F=f(u,v)\sqrt{\frac{D(0)}{D(s)}}\cdot e^{ik(\psi_0+s)}. \tag{3.08} \]
The quantities \(u, v, s\) may be interpreted as curvilinear coordinates of a point in space, connected with the rectangular coordinates \(x, y, z\) by relations (2.11). The geometrical meaning of these curvilinear coordinates is obvious: the parameters \(u, v\) determine the position of that point on the surface of the body from which the ray that has reached the point \(x, y, z\) was reflected; the quantity \(s\) is the distance traversed by the ray after reflection.
Thus, the quantity \(F\) in formula (3.08) may be interpreted as a function of a point in space. It is evident that on the surface \(s=0\) this function takes the value (3.06). It is also evident that it satisfies the radiation condition and corresponds to the scattered wave. But, in addition, if the parameter \(k\) is large, the function \(F\) approximately satisfies the wave equation. Indeed, it can be shown that from the definition of \(\psi_0\) and \(D(s)\) and from formulas (2.11) the equalities follow
\[ \{\operatorname{grad}(\psi_0+s)\}^2=1, \tag{3.09} \]
\[ \operatorname{div}\left\{f^2\frac{D(0)}{D(s)}\operatorname{grad}(\psi_0+s)\right\}=0. \tag{3.10} \]
On the basis of these equalities it is easy to verify that, after substituting \(F\) into equation (3.07), the terms of second and
of the first degree with respect to \(k\), and only the terms of zero degree will remain.
Independently of the reasoning just presented, the validity of expression (3.08) follows from considerations of geometrical optics. Indeed, this expression must give the reflected wave. But the phase of the reflected wave is, evidently, equal to \(k(\psi_0+s)\). As for the amplitude, if one proceeds along a narrow pencil of reflected rays, the amplitude must vary inversely proportionally to the square root of the cross section of the pencil, which is precisely what is given by formula (3.08).
Thus this formula gives the field of the reflected wave at a distance \(s\) from the surface when the field on the surface itself is known.
Applying the formula found to the components of the electric and magnetic fields, we obtain for them the expressions
\[ \mathbf{E}=\mathbf{E}^{*}(u,v)\cdot \sqrt{\frac{D(0)}{D(s)}}\, e^{ik(\psi_0+s)}, \tag{3.11} \]
\[ \mathbf{H}=\mathbf{H}^{*}(u,v)\cdot \sqrt{\frac{D(0)}{D(s)}}\, e^{ik(\psi_0+s)}, \tag{3.12} \]
where \(\mathbf{E}^{*}(u,v)\) and \(\mathbf{H}^{*}(u,v)\) are the field amplitudes on the surface of the body obtained from Fresnel’s formulas.
The formulas obtained by us for the field represent a natural combination of the laws of reflection and of geometrical (ray) optics. Each of these separately had been known for more than a hundred years: Fresnel found his laws of reflection around 1820, and Hamilton the laws of ray optics around 1830. In particular, Hamilton knew that the quantity corresponding to our \(D(s)\) is a polynomial of the second degree in \(s\). However, we have not been able to find in the literature any indication of the application of these results to an approximate representation of the reflected electromagnetic wave.
4. THE LAW OF DIFFRACTION IN THE REGION OF THE PENUMBRA
We have already mentioned in the introduction that near the geometrical boundary of the shadow, in the region of grazing incidence of the ray, the incident and reflected waves become inseparable from one another and Fresnel’s formulas become inapplicable. We shall present here, on the basis of our work1, the idea of deriving diffraction formulas that give the field in this region, as well as in the region of the penumbra and shadow.
Let us imagine a convex body on which a plane wave is incident in the direction of the \(x\)-axis. Choose on the surface of the body a point lying on the boundary of the geometrical shadow and take it as the origin of coordinates. Direct the \(z\)-axis along the normal to the surface (toward the air). Since the normal at the boundary of the shadow is perpendicular to the direction of the wave, our \(x\)- and \(z\)-axes will be mutually perpendicular. We choose the \(y\)-axis so as to obtain a right-handed coordinate system.
In a neighborhood of the given point the equation of the surface will have the form
\[ z+\frac{1}{2}(ax^2+2bxy+cy^2)=0, \tag{4,01} \]
where
\[ a\geqslant 0;\qquad c\geqslant 0;\qquad ac-b^2\geqslant 0. \tag{4,02} \]
The radius of curvature of the normal section of the surface will be equal to
\[ R_0=\frac{1}{a}. \tag{4,03} \]
In what follows we shall introduce the “large parameter” \(m\) by the formula
\[ m=\sqrt[3]{\frac{kR_0}{2}}=\sqrt[3]{\frac{k}{2a}} \tag{4,04} \]
and shall solve our problem, neglecting quantities of order \(\dfrac{1}{m^2}\) in comparison with unity.
Our aim is to find the electromagnetic field at distances from the origin that are small in comparison with the radius of curvature \(R_0\).
Under our assumptions each component of the field will be of the form
\[ F=e^{ikx}F^*, \tag{4,05} \]
where \(F^*\) satisfies the differential equation
\[ \frac{\partial^2 F^*}{\partial z^2}+2ik\,\frac{\partial F^*}{\partial x}=0. \tag{4,06} \]
All components of the field can be expressed in terms of \(H_y\) and \(H_z\) by the formulas
\[ \begin{aligned} E_x&=-\frac{i}{k}\left(\frac{\partial H_z}{\partial y}-\frac{\partial H_y}{\partial z}\right),\\ E_y&=H_z,\\ E_z&=-H_y,\\ H_x&=\frac{i}{k}\left(\frac{\partial H_y}{\partial y}+\frac{\partial H_z}{\partial z}\right), \end{aligned} \tag{4,07} \]
which may be regarded as simplified Maxwell equations.
Approximate limiting conditions for the field in air at the boundary with a good conductor were established by M. A. Leontovich. They are valid under the conditions
\[ |\eta|\gg 1;\qquad kR_0\sqrt{|\eta|}\gg 1 \tag{4,08} \]
and have the form (see (1,18)).
\[ \mathbf{E}-\mathbf{n}E_n=\sqrt{\frac{\mu}{\eta}}\,[\mathbf{n}\times\mathbf{H}]. \tag{4,09} \]
In what follows we shall take \(\mu=1\). The components of the normal vector entering into (4,09) are determined from the equation of the surface (4,01). We may approximately put
\[ n_x=ax+by;\qquad n_y=bx+cy;\qquad n_z=1, \tag{4,10} \]
for the squares of the quantities \(n_x\) and \(n_y\) may be neglected in comparison with unity. We shall regard the quantities \(n_x, n_y, \dfrac{1}{m}, \dfrac{1}{\sqrt{\eta}}\) as small quantities of the same order.
Under these assumptions one can derive from (4,07) and (4,09) the following boundary conditions for the field, containing only \(H_y\) and \(H_z\). They will have the form:
\[ H_z=-n_y H_y, \tag{4,11} \]
\[ \frac{\partial H_y}{\partial z} +ik\left(n_x+\frac{1}{\sqrt{\eta}}\right)H_y = n_y\frac{\partial H_z}{\partial x}. \tag{4,12} \]
Owing to the smallness of the quantity \(n_y\), the right-hand sides of these equations represent correction terms. In the first approximation they may be replaced by zero and the simpler boundary conditions may be considered:
\[ H_z=0, \tag{4,13} \]
\[ \frac{\partial H_y}{\partial z} +ik\left(n_x+\frac{1}{\sqrt{\eta}}\right)H_y=0. \tag{4,14} \]
In the second approximation, one may substitute into the right-hand sides of (4,11) and (4,12) the values of \(H_y\) and \(H_z\) obtained by solving the differential equations with the boundary conditions (4,13) and (4,14)*).
In addition to the differential equations and the boundary conditions on the surface of the body, the solution must satisfy conditions at infinity. These latter consist in the requirement that that part of the solution which corresponds to a plane wave have the prescribed amplitude at infinity.
The mathematical problem thus posed has a unique solution, which we shall give here, omitting all derivations and confining ourselves to definitions.
If the factor \(e^{ikx}\) is not counted, the field will depend on the coordinates only through the quantities**)
\[ \xi=m(ax+by), \tag{4,15} \]
\[ \zeta=2am^2\left[z+\frac{1}{2}(ax^2+2bxy+cz^2)\right], \tag{4,16} \]
*) In our principal work\(^1\) an inconsistency was admitted here; namely, (4,11) and (4,14) were taken as the boundary conditions. As a result, in the final expression for \(H_z\) (see below formula (4,30)) both the principal term and a correction term were obtained, while in the expression for \(H_y\), formula (4,29), only the principal term was obtained. In the present article this inaccuracy has been corrected.
**) The correction terms will, in addition, contain the coordinate \(y\) linearly.
of which the second vanishes on the surface. The constants characterizing the electrical properties of the reflecting surface enter into the expressions for the field by means of the quantity*)
\[ q=\frac{im}{\sqrt{\eta}};\qquad m=\sqrt[3]{\frac{k}{2a}}. \tag{4,17} \]
The field is ultimately expressed through a single universal (i.e., independent of the shape of the surface) function \(V_1(\xi,\zeta,q)\) and through its limiting value
\[ V_2(\xi,\zeta)=V_1(\xi,\zeta,\infty). \tag{4,18} \]
The function \(V_1\) can be represented in the form of a definite integral containing the complex Airy functions \(w_1(t)\) and \(w_2(t)\). The latter are defined as solutions of the differential equation
\[ w''(t)=tw(t), \tag{4,19} \]
which have, for large negative \(t\), the asymptotic expressions
\[ w_1(t)=\frac{1}{\sqrt[4]{-t}}\exp\left(i\frac{2}{3}(-t)^{3/2}+i\frac{\pi}{4}\right), \tag{4,20} \]
\[ w_2(t)=\frac{1}{\sqrt[4]{-t}}\exp\left(-i\frac{2}{3}(-t)^{3/2}-i\frac{\pi}{4}\right). \tag{4,21} \]
The expression for \(V_1\) has the form
\[ V_1(\xi,\zeta,q)=\frac{i}{2\sqrt{\pi}}\int_C e^{i\xi t} \left\{w_2(t-\zeta)- \frac{w'_2(t)-qw_2(t)}{w'_1(t)-qw_1(t)}\,w_1(t-\zeta)\right\}\,dt, \tag{4,22} \]
where the contour \(C\) goes along the ray \(\arg t=\frac{2}{3}\pi\) from infinity to zero and along the ray \(\arg t=-\frac{1}{3}\pi\) from zero to infinity.
For \(\zeta=0\) (on the surface of the body) the expression for \(V_1\) is simplified and assumes the form
\[ V_1(\xi,0,q)=\frac{1}{\sqrt{\pi}}\int_C e^{i\xi t}\frac{dt}{w'_1(t)-qw_1(t)}. \tag{4,23} \]
This function has been tabulated for a number of values of \(q\); the tables for \(q=0\) (an absolutely conducting body) were printed in our paper \(^{2}\).
*) If, instead of the Leontovich conditions (1,18), we had used formulas (1,19) and (1,20), we would have obtained for \(q\) a somewhat more exact value
\[ q=\frac{im}{\eta}\sqrt{\eta-1}. \]
Having the definition of \(V_1(\xi,\zeta,q)\), we can write expressions for the fields. For this we introduce the functions
\[ \Psi=e^{-i\varphi}V_1(\xi,\zeta,q), \tag{4.24} \]
\[ \Phi=e^{-i\varphi}V_2(\xi,\zeta), \tag{4.25} \]
where
\[ \varphi=\xi\zeta-\frac{1}{3}\xi^3, \tag{4.26} \]
and, with their aid, form the expressions:
\[ P=-i\frac{b}{a}\frac{\partial\Psi}{\partial\zeta} +\left(i\frac{b}{a}q+\frac{ac-b^2}{a}my\right)(\Phi-\Psi), \tag{4.27} \]
\[ Q=i\frac{b}{a}\frac{\partial\Phi}{\partial\zeta} +\left(-i\frac{b}{a}q+\frac{ac-b^2}{a}my\right)(\Phi-\Psi). \tag{4.28} \]
Then the components \(H_y\) and \(H_z\) of the magnetic field will be equal to:
\[ H_y=H_y^0 e^{ikx}\Psi+\frac{1}{m}H_z^0 e^{ikx}Q, \tag{4.29} \]
\[ H_z=\frac{1}{m}H_y^0 e^{ikx}P+H_z^0 e^{ikx}\Phi, \tag{4.30} \]
where \(H_y^0\) and \(H_z^0\) are the amplitudes of the incident wave. All four functions \(\Phi,\ \Psi,\ P,\ Q\) satisfy a differential equation of the form (4.06) and will be of the same order of magnitude. Since \(m\) is a large parameter, the terms containing \(\Phi\) and \(\Psi\) will be the principal terms, while the terms containing \(P\) and \(Q\) will be correction terms. The field components \(E_x\) and \(H_x\) will be of the same order as the correction terms, namely:
\[ E_x=-\frac{i}{m}H_y^0 e^{ikx}\frac{\partial\Psi}{\partial\zeta}, \tag{4.31} \]
\[ H_x=\frac{i}{m}H_z^0 e^{ikx}\frac{\partial\Phi}{\partial\zeta}. \tag{4.32} \]
As for the remaining components of the electric field, by virtue of the simplified Maxwell equations (4.07) they will be equal to
\[ E_y=H_z;\qquad E_z=-H_y. \tag{4.33} \]
Thus, all components of the field have been determined by us.
5. INVESTIGATION OF THE EXPRESSIONS FOR THE FIELD IN THE SHADOW AND ILLUMINATED REGIONS
The diffraction formulas derived by us give the field near a certain point lying on the surface of the conducting body at the boundary of the geometrical shadow. We shall show that they give a continuous transition from the field corresponding to the Fresnel formulas (for the illuminated region) to complete shadow. Let us begin with the shadow region.
The integral (4.22) can be represented as a sum of residues corresponding to the roots of the denominator of the integrand. We have
\[ V_1(\xi,\zeta,q)= i\,2\sqrt{\pi}\sum_{s=1}^{\infty} e^{i\xi t_s}\, \frac{w_1(t_s-\zeta)}{w_1'(t_s)(t_s-q^2)}, \tag{5,01} \]
where \(t_s\) is a root of the equation
\[ w_1'(t_s)-q w_1(t_s)=0. \tag{5,02} \]
The roots \(t_s\) lie near the ray \(\arg t=\dfrac{\pi}{3}\) and increase in modulus.
For sufficiently large positive values of \(\xi-\sqrt{\zeta}\) in the series (5,01) one may restrict oneself to a single term. If, moreover, one uses the asymptotic expression (4,20) for \(w_1\) and regards \(\zeta\) in it as large in comparison with \(t_1\), we obtain for \(V_1\) the approximate expression
\[ V_1(\xi,\zeta,q)= \frac{e^{i\frac{3\pi}{4}}\,2\sqrt{\pi}}{w_1'(t_1)(t_1-q^2)} \cdot e^{i\frac{2}{3}\zeta^{3/2}}\cdot e^{i(\xi-\sqrt{\zeta})t_1}. \tag{5,03} \]
The quantity \(t_1\) has, for \(q=0\) and \(q=\infty\), the following values:
\[ t_1=1.01879\cdot e^{i\frac{\pi}{3}} \qquad (q=0), \tag{5,04} \]
\[ t_1=2.33811\cdot e^{i\frac{\pi}{3}} \qquad (q=\infty). \tag{5,05} \]
In any case, both the real and the imaginary parts of \(t_1\) are positive. Hence it follows that, as \(\xi-\sqrt{\zeta}\) increases, the functions \(V_1\) and \(V_2\) and the functions \(\Phi, \Psi, P, Q\) connected with them, and consequently the field, will decrease according to an exponential law.
Let us note that the equality \(\xi-\sqrt{\zeta}=0\) gives the geometrical boundary of the shadow. Increasing positive values of the quantity \(\xi-\sqrt{\zeta}\) correspond to points lying farther and farther in the region of shadow.
Where the quantity \(\xi-\sqrt{\zeta}\) is not large (it may be of either sign), we have the penumbra region. We shall not dwell on methods for calculating the function \(V_1\) in this region; we shall indicate only that this function, and consequently also the field, vary there in a continuous manner.
Let us now pass to the illuminated region, where the quantity \(\xi-\sqrt{\zeta}\) is large and negative. In this case the series (5,01) for \(V_1\) cannot be used and it is necessary to return to the integral (4,22). The term containing \(w_2(t-\zeta)\) in this integral can be evaluated exactly. It gives
\[ \frac{i}{2\sqrt{\pi}}\int_C e^{i\xi t} w_2(t-\zeta)\,dt=e^{i\varphi}, \tag{5,06} \]
where \(\varphi\) has the value
\[ \varphi=\xi\zeta-\frac{1}{3}\xi^3, \tag{5,07} \]
coinciding with (4,26). Thus, this term gives in the functions \(\Psi\) and \(\Phi\) a term equal to unity, while in the expressions for the field it corresponds to the incident wave.
The second term may be calculated by the method of stationary phase, as shown in our paper \(^{1}\). The extremum of the phase is obtained for \(\sqrt{-t}=p\), where
\[ p=\frac{1}{3}\left(\sqrt{\xi^2+3\zeta}-2\xi\right). \tag{5,08} \]
For the square root entering this formula it is convenient to introduce a special notation:
\[ \sigma=\sqrt{\xi^2+3\zeta}. \tag{5,09} \]
We note that the quantity \(p\) has the same sign as \(\sqrt{\zeta}-\xi\), so that \(p>0\) corresponds to the illuminated region, \(p=0\) to the geometrical boundary of the shadow, and \(p<0\) to the shadow region. We are now interested in large positive values of \(p\). For this case the application of the stationary-phase method gives, for the whole quantity \(V_1\), the expression
\[ V_1(\xi,\zeta,q)=e^{i\varphi}-e^{i\varphi^*}\sqrt{\frac{p}{\sigma}}\cdot\frac{q-ip}{q+ip}, \tag{5,10} \]
where the phase \(\varphi\) is equal to (5,07), and the phase \(\varphi^*\) is equal to
\[ \varphi^*=\frac{1}{27}\left(4\sigma^3-3\sigma^2\xi-2\xi^3\right). \tag{5,11} \]
We note that the phase difference \(\varphi^*-\varphi\) is equal to
\[ \varphi^*-\varphi=\frac{2}{27}(\sigma+\xi)(\sigma-2\xi)^2=(\sigma-p)p^2. \tag{5,12} \]
For \(\zeta=0\) we shall have \(\sigma=p=-\xi\), so that \(\varphi^*-\varphi\) vanishes on the surface of the body.
The quantity \(V_2\) is obtained from (5,10) for \(q=\infty\). The functions \(\Psi\) and \(\Phi\) connected with \(V_1\) and \(V_2\) will be approximately equal to:
\[ \Psi=1-e^{i(\varphi^*-\varphi)}\sqrt{\frac{p}{\sigma}}\cdot\frac{q-ip}{q+ip}, \tag{5,13} \]
\[ \Phi=1-e^{i(\varphi^*-\varphi)}\sqrt{\frac{p}{\sigma}}. \tag{5,14} \]
The expressions for the field contain not only the functions \(\Psi\) and \(\Phi\) themselves, but also their derivatives with respect to \(\zeta\). In forming the derivatives all factors, except the phase factor, may be regarded as constant. Consequently
\[ \frac{\partial(\varphi^*-\varphi)}{\partial\zeta} =\frac{2}{3}\sigma-\frac{4}{3}\xi=2p \tag{5,15} \]
we shall have
\[ \frac{\partial \Psi}{\partial \zeta}=2ip(\Psi-1);\quad \frac{\partial \Phi}{\partial \zeta}=2ip(\Phi-1). \tag{5,16} \]
Computing, with the aid of these values, the quantities \(P\) and \(Q\), we obtain
\[ P=Q=\frac{2ip}{q+ip}\left(\frac{b}{a}p-\frac{ac-b^{2}}{a}my\right)\sqrt{\frac{p}{\sigma}}\,e^{i(\varphi^{*}-\varphi)}. \tag{5,17} \]
It remains for us to substitute the expressions found into formulas (4,29)—(4,32) for the field. In doing this it is convenient to denote by a single letter
\[ \chi=kx+\varphi^{*}-\varphi \tag{5,18} \]
the phase of the reflected wave. With this notation we shall have
\[ H_y=H_y^0e^{ikx}-H_y^0\frac{q-ip}{q+ip}\sqrt{\frac{p}{\sigma}}\,e^{i\chi}+ \]
\[ +\frac{1}{m}H_z^0\frac{2ip}{q+ip}\left(\frac{b}{a}p-\frac{ac-b^{2}}{a}my\right)\sqrt{\frac{p}{\sigma}}\,e^{i\chi}, \tag{5,19} \]
\[ H_z=H_z^0e^{ikx}-H_z^0\sqrt{\frac{p}{\sigma}}\,e^{i\chi}+ \]
\[ +\frac{1}{m}H_y^0\frac{2ip}{q+ip}\left(\frac{b}{a}p-\frac{ac-b^{2}}{a}my\right)\sqrt{\frac{p}{\sigma}}\,e^{i\chi}. \tag{5,20} \]
\[ E_x=-\frac{1}{m}H_y^0\cdot 2p\,\frac{q-ip}{q+ip}\sqrt{\frac{p}{\sigma}}\,e^{i\chi}, \tag{5,21} \]
\[ H_x=\frac{1}{m}H_z^0\cdot 2p\sqrt{\frac{p}{\sigma}}\,e^{i\chi} \tag{5,22} \]
and, furthermore, \(E_y=H_z;\ E_z=-H_y\).
The first terms in (5,19) and (5,20), obviously, give the incident wave, while the remaining terms give the reflected wave. In the following paragraph we shall show that the reflected wave exactly corresponds to the Fresnel formulas with a correction for beam divergence.
6. COMPARISON OF THE DIFFRACTION FORMULAS WITH THE FRESNEL FORMULAS FOR THE ILLUMINATED REGION
Let us now turn to the Fresnel formulas. Putting \(\mu=1\) in the Fresnel coefficients and considering \(\sqrt{\eta}\) to be a large quantity, and \(\cos\vartheta\) small (of order \(1/\sqrt{\eta}\)), we obtain for \(N\) and \(M\) the expressions
\[ N=\frac{\sqrt{\eta}\cos\vartheta-1}{\sqrt{\eta}\cos\vartheta+1};\quad M=-1 \tag{6,01} \]
In the Fresnel formulas (1,13) and (1,14) we must put \(a_x=1\), \(a_y=a_z=0\) and regard \(n_x\) and \(n_y\) as small quantities, the squares of which—
which can be neglected. These formulas then give, for the electric field,
\[ \left. \begin{aligned} E_x^*&=-2Nn_xH_y^0,\\ E_y^*&=-H_z^0-(N+1)n_yH_y^0,\\ E_z^*&=-NH_y^0+(N+1)n_yH_z^0 \end{aligned} \right\} \tag{6,02} \]
and, for the magnetic field,
\[ \left. \begin{aligned} H_x^*&=-2n_xH_z^0,\\ H_y^*&=NH_y^0-(N+1)n_yH_z^0,\\ H_z^*&=-H_z^0-(N+1)n_yH_y^0. \end{aligned} \right\} \tag{6,03} \]
To obtain the field of the reflected wave at some distance from the surface, it is necessary, according to (3,11) and (3,12), to multiply these expressions by the factor
\[ \sqrt{\frac{D(0)}{D(s)}}\,e^{ik(x_0+s)} . \tag{6,04} \]
The values of all quantities, except \(s\), must be taken at that point \(x_0, y_0, z_0\) where the reflection occurred of the ray arriving at the point \(x, y, z\). Since the equation of the reflecting surface is
\[ z_0+\frac{1}{2}(ax_0^2+2bx_0y_0+cy_0^2)=0, \tag{6,05} \]
we have
\[ n_x=ax_0+by_0;\qquad n_y=bx_0+cy_0;\qquad n_z=1. \tag{6,06} \]
In computing \(D(s)\) from the general formula (2,23), we must neglect the last term, since we are interested in the field at distances small in comparison with the radii of curvature. The remaining terms give
\[ D(s)=\cos\vartheta+2as=2as-ax_0-by_0. \tag{6,07} \]
In order to compare the diffraction formulas (5,19)—(5,22) with the Fresnel formulas (6,02) and (6,03), we must establish the relation between the quantities \(x_0, y_0, s\) and the coordinates \(x, y, z\) (or the quantities \(\xi, \zeta, y\)). This relation is given by formulas (2,11), which in our case take the form:
\[ \left. \begin{aligned} x&=x_0+s-2sn_x^2,\\ y&=y_0-2sn_xn_y,\\ z&=z_0-2sn_xn_z. \end{aligned} \right\} \tag{6,08} \]
Solving these equations approximately with respect to \(x_0,\ y_0,\ s\), we obtain
\[ \begin{aligned} a x_0 + b y_0 &= \frac{2\xi-\sigma}{3m}=-\frac{p}{m},\\ y_0&=y,\\ s&=\frac{\sigma+\xi}{3am}=\frac{\sigma-p}{2am}. \end{aligned} \tag{6,09} \]
Hence
\[ n_x=-\frac{p}{m};\qquad n_y=-\frac{b}{a}\frac{p}{m}+\frac{ac-b^2}{a}\,y . \tag{6,10} \]
Next, the phase \(\chi\), by definition (5,12), (5,18), is equal to
\[ \begin{aligned} \chi&=kx+\varphi^*-\varphi=kx+(\sigma-p)p^2=\\ &=k\left(x+2s n_x^2\right)=k(x_0+s), \end{aligned} \tag{6,11} \]
i.e. it is equal to the phase of the reflected wave computed by geometrical optics. Let us now compute the quantity \(D(s)\). Substituting the quantities (6,09) into (6,07), we obtain
\[ D(s)=\frac{\sigma}{m}, \tag{6,12} \]
where, obviously,
\[ D(0)=\cos\vartheta=\frac{p}{m}. \tag{6,13} \]
The last three formulas give
\[ \sqrt{\frac{p}{\sigma}}\,e^{i\chi} = \sqrt{\frac{D(0)}{D(s)}}\,e^{ik(x_0+s)} . \tag{6,14} \]
Thus the factor (6,14), entering into all expressions for the reflected wave in the diffraction formulas (5,19)—(5,22), coincides with the factor entering into formulas (3,08)—(3,09), which represent a generalization of Fresnel’s formulas. The quantity
\[ \sqrt{\frac{p}{\sigma}} = \sqrt{\frac{1}{3}-\frac{2\xi}{3\sigma}} \tag{6,15} \]
then gives the correction for the spreading of the beam.
It remains for us to verify that all the other quantities in formulas (5,19)—(5,22) also coincide with the Fresnel ones.
According to (4,17) and (6,13) we have
\[ q=\frac{im}{\sqrt{\eta}};\qquad p=m\cos\vartheta. \tag{6,16} \]
Therefore
\[ \frac{q-ip}{q+ip}=\frac{1-\sqrt{\eta}\cos\vartheta}{1+\sqrt{\eta}\cos\vartheta}=-N, \tag{6,17} \]
where \(N\) is the Fresnel coefficient (6,01)\(^*\).
Using formulas (6,10) and (6,17) as notation, we can write our expressions (5,19)—(5,22) for the field in the form
\[ H_y=H_y^0 e^{ikx}+\left[NH_y^0-(N+1)n_yH_z^0\right]\sqrt{\frac{p}{\sigma}}\,e^{il}, \tag{6,18} \]
\[ H_z=H_z^0 e^{ikx}+\left[-H_z^0-(N+1)n_yH_y^0\right]\sqrt{\frac{p}{\sigma}}\,e^{il}, \tag{6,19} \]
\[ E_x=-2Nn_xH_y^0\sqrt{\frac{p}{\sigma}}\,e^{il}, \tag{6,20} \]
\[ H_x=-2n_xH_z^0\sqrt{\frac{p}{\sigma}}\,e^{il}. \tag{6,21} \]
Comparing these expressions with the Fresnel formulas (6,02) and (6,03), we see that the factors for the quantities (6,14) coincide exactly with their Fresnel values \(H_y^*, H_z^*, E_x^*, H_x^*\). The equalities \(E_y=H_z;\ E_z=-H_y\) are satisfied both in the case of our formulas and in the case of Fresnel’s formulas.
Thus, we have shown that in that part of the illuminated region where the angle of inclination of the ray to the surface of the body is small, our formulas pass into the generalized Fresnel formulas (by introducing the factor (6,14)).
In the penumbral and shadow regions, however, our formulas give the diffraction pattern.
CITED LITERATURE
- V. A. Fock, Izv. AN SSSR, ser. fiz. 10, No. 2, pp. 171—186 (1946).
- V. A. Fock, ZhETF 15, issue 12, pp. 693—702 (1945).
\[ \text{*) The value } q=\frac{im}{\eta}\sqrt{\eta-1} \text{ leads to a somewhat more accurate value of } N,\text{ namely} \]
\[ N=\frac{\eta\cos\vartheta-\sqrt{\eta-1}}{\eta\cos\vartheta+\sqrt{\eta-1}}. \]