2.3 Geometrical opticsAdvanced theoryToday&s geometrical optics (GO) enables to examine the propagation
of electromagnetic waves even in complex inhomogeneous media of a given type. As the classical
geometrical optics, GO is based on the notion that waves propagate along
beams,
which can be even curvilinear now. GO is based on the exact mathematical theory coming out
Maxwell equations and enabling to compute not only wave trajectories but too variation of
magnitude and phase of field intensity and wave polarization.
Moreover, the method is well-understandable thanks to the simple
notion of beams.
As already described in the layer A, theory of geometrical
optics is based on the following two assumptions:
-
Wave number
k is high (so that some terms containing the higher power of k in the denominator can be neglected during mathematical rearrangements). This assumption predetermines geometrical
optics to the applications in the area of optical frequencies. Since the wave number is not
needed being extremely high, the method can be used even for radio frequencies.
- The wave is examined in a relatively long distance from the source. In such distance, wave amplitude varies slowly whereas the wave phase changes quickly
(for 2π for every wavelength).
In the following, our attention is focused in the media exhibiting
a continuous variation permittivity and permeability. The media are assumed to be
lossless. Refractive index of the medium
is a real number and is a function of the position
(x, y, z). Symbol s denotes the
coordinate (curvilinear in general), which is measured along the direction of propagation.
Investigating the wave propagation in a homogeneous medium,
wave number
k = k0(εrμr)1/2 = k0n and the propagation direction are
usually known. E.g., the following relation is valid for the field intensity of a
plane wave
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(s is in the propagation direction)
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( 2.3B.2a )
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On the part of the trajectory ds,
the phase changes for
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.
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( 2.3B.2b )
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In the coordinate system x, y, z, the propagation direction need not to be identical
with the direction of a coordinate axis. We usually solve this situation by considering
the wave number
k as a vector oriented to the propagation direction. Its components kx, ky, kz,
to the directions of coordinate axes are wave numbers in the directions x, y, z. In two arbitrarily located points
A, B, which coordinates differ for dx, dy, dz, the phases differ for
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.
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( 2.3B.3 )
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Geometrical optics does not suppose any a priori knowledge of the
direction of wave propagation, because it is different in different points (in the medium of
a variable refractive index).
In parallel with (2.3B.2), we use
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( 2.3B.4a )
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where the distance is not explicitly present. The quantity L
is a scalar function of coordinates x, y, z,
it is called eiconale and
it determines the phase of field intensity in every point. Comparing (2.3A.4b) and (2.3A.2b) shows that
on the infinitely short path ds (where n = const), is
Considering (2.3A.4), equation
L = const is the equation of a plane, where wave
is of the same phase, i.e. the equation of a wave surface.
Vector grad L is of the direction of the steepest variation of
the eiconale (i.e., even of the phase), and this is the direction of wave propagation.
For example:
and according to (2.3A.5) is
and therefore
or
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( 2.3B.7 )
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As already said in the layer A, the equation
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( 2.3B.8 )
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is considered as the basic equation of the geometrical optics.
The vector grad L is of the direction of wave propagation
in every point, and the curve, which tangent is of the direction grad L
in every point, is called beam.
Beams can be curvilinear and eqn. (2.3B.8) is the differential equation of beams.
For practical calculations, the form (2.3B.8) is inconvenient.
Instead, following equations are used:
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,
,
,
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( 2.3B.9 )
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.
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( 2.3B.10 )
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As already said, s is the
coordinate along the beam. Now, the way of deriving the
first of relations (2.3B.9) is demonstrated.
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Fig. 2.3B.1 | Two-dimensional inhomogenous medium |
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In a homogeneous medium, we exploit in computations
the wave number, and we understand it as a
vector, which is oriented to the propagation direction. Components of k are
kx
and ky
(in two dimensions) and we can imagine that wave propagates from
any point along the pointed trajectory going in parallel to x (wave number kx
)
and then, going in parallel to y (wave number ky
). n an inhomogeneous medium,
the same approach can be used within a very small region dx, dy only, where
the refractive index
can be considered constant. In the frame of the whole medium, kx and ky
vary, and those variations have to be examined first.
In an inhomogeneous medium, computing with refractive index
is preferred to computing with wave number. Since k = k0 n
and wave number is a vector, even the right-hand side of the equation has to be a vector.
Hence, the refractive index n assumed being vector. Its magnitude is
n and its direction is identical with the wave propagation (in general, different
in different points). Components nx and ny correspond to
respective wave numbers, e.g. kx = k0 nx. This formalism converted the inhomogeneous medium to the
anisotropic medium, which is not truth from
the microscopic point of view. From the macroscopic point of view, this might be considered
as a truth: if wave enters the medium in various directions, the trajectory is incurvated
different way every time. In the homogeneous medium, the situation is different.
Assume a two-dimensional medium, where
refractive index varies in the direction x only.
In fig. 2.3B.1, two elementary layers of this medium are depicted, including respective quantities.
Let us examine variations of components of refractive index nx = n cos(α) and ny = nsin(α) in the direction x:
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,
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Rewriting goniometric functions, multiplying, and substituting sin (dα) = dα, cos(dα) = 1 yield
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,
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( 2.3B.11a )
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.
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( 2.3B.11b )
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Next, the variation of dny
in the direction x,
i.e. dny/dx, has to equal zero. This is caused by the fact that two waves
propagating along the boundary at different sides near each other have to be of the same
wave number, although they propagate in different media.
In the opposite case, a continuously rising phase shift between these two waves would exists, and on the
boundary, a phase jump would appear, which is impossible.
In (2.3B.11b), we put dny = 0, express dα and substitute to (2.3B.11a). We get:
The above relation is rather interesting. It shows that the variation dnx
is higher than the variation of the refractive index dn at the same point.
The next steps are relatively simple. Eqn. (2.3B.12) is divided by dx
and rearranged: dnx/dx = (dnx/ds)(ds/dx) = (dnx/ds)(1/cos(α)).
The term cos(α) vanishes and we get the relation (2.3B.9a).
Exploitation of (2.3B.9) and (2.3B.10) is demonstrated by the following example.
Assume a two-dimensional medium (x, y), where the permittivity ε, and consequently the
refractive index
n, continuously vary along the coordinate y in accordance with the function n(y); along the coordinate x,
the refractive index is constant. Since ∂n/∂x = 0, eqn. (2.3B.9a) results to:
Here, C is an integration constant. Substituting to (2.3B.10),
the term containing z is deleted because the task is two-dimensional, and we get
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( 2.3B.14 )
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Dividing both equations, the variable s
is eliminated and the differential equation of the beam is obtained:
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( 2.3B.15 )
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We substitute a given function n(y) and integrated (analytically or numerically).
The equation of the beam y = f(x) is the result.
The integration constant C is determined using a
boundary condition, a known
beam
direction in a given point. If the value of the eiconale is known in this point LA (A e.g.),
we can compute its value in another different point (B e.g.) on the same beam. Using (2.3B.5), we get
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( 2.3B.16 )
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if the integration path follows the beam. In our task, the
differential ds is replaced by the differential dy
according to (2.3B.14), n is replaced by n(y) and integration is performed
according toy from yA
to yB
.
[End of example]
In layer A, we have found that
geometrical optics enables to compute intensity changes during the
propagation. This ability is conditioned by the knowledge of computing the shape of
beams, i.e. the trajectory of wave.
Computation of the intensity is based on the fact that wave energy propagates along beams. This means, that no electromagnetic energy leaves or enters the walls of a tube, which appears when displaying many neighboring beams and which
transversal cut gives a closed curve. Energy propagates inside the tube along beams only.
On the basis of this idea, the following relation was derived in the layer A:
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( 2.3B.17 )
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Fig. 2.3B.2 | Astigmatic beam |
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Eqn. (2.3B.17) is not valid for regions, where
beams cut (which would cause an infinite field intensity). Such situation appears in a focus and on a
caustics.
The term
caustics is explained thereinafter.
Sometimes, beams can exhibit different curvature radii of
wave surface in different planes (e.g., astigmatic beams).
A part of such a tube is depicted in fig. 2.3B.2.
The curvature center of the equiphase surface in the horizontal plane
is denoted as Oh, and in the vertical one as Ov. The distances of facet dS1 and Oh (Ov) is r1 (r2),
r is the distance between facets dS1 and dS2.
Considering fig. 2.3B.2, we get:
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, .
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Dividing relations, we express the radius of both the facets, and
using (2.3B.17), we compute the ratio of intensities. For a homogeneous medium, when respecting
phase variation, we get:
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( 2.3B.18 )
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The described method of the geometric optics can be used to the examination of the electromagnetic wave propagation through the medium which structure
corresponds to the idealized
ionospheric layer
(if the scaling factor is not considered). Wave radiated by a
point source
propagates through the medium with the unitary dielectric constant first, and then continues through the layer which dielectric constant decreases down
to a certain minimum, and then rises to 1 again. For the given dependency of the dielectric constant in the layer and for the given position of a source,
the wave trajectory, the
equiphase surface
and the
eiconale
increase can be computed by numeric integration.
Below the layer, where several waves are propagating (due to the trajectory curvature), the direction of the propagation, field intensity and
eiconale
(phase) can be computed for each wave. Computations show that beams return back by nice curls in a certain part of space. Tops of the curls can be
connected by an imaginary spline, which is not crossed by any beam, and in a close vicinity of which (below) two neighboring beams intersect. This
spline (or a surface in space) is called
caustics.
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