A generalized form of Snell's law in anisotropic media

Abstract

We have reformulated the law governing the refraction of rays at a planar interface separating two anisotropic
media in terms of slowness surfaces. Equations connecting ray directions and phase-slowness angles are derived using geometrical properties of the gradient operator in slowness space. A numerical example shows that, even in weakly anisotropic media, the ray trajectory governed by the anisotropic Snell’s law is significantly different from that obtained using the isotropic
form. This could have important implications for such considerations as imaging (e.g., migration) and lithology analysis (e.g., amplitude variation with offset).
Expressions are shown specifically for compressional (qP) waves but they can easily be extended to SH waves by equating the anisotropic parameters (i.e., ε=δ⇒γ ) and to qSV and converted waves by similar means. The analytic expressions presented are more complicated than the standard form of Snell’s law. To facilitate practical application, we include our Mathematica code.

Full text

GEOPHYSICS, VOL. 65, NO. 2 (MARCH-APRIL 2000); P. 632–637, 4 FIGS., 1 TABLE.
A generalized form of Snell’s law in anisotropic media
Michael A. Slawinski
∗
, Rapha¨el A. Slawinski
‡
, R. James Brown
‡
,
and John M. Parkin
∗∗
ABSTRACT
We have reformulated the law governing the refrac-
tion of rays at a planar interface separating two aniso-
tropic media in terms of slowness surfaces. Equations
connecting ray directions and phase-slowness angles are
derived using geometrical properties of the gradient op-
erator in slowness space. A numerical example shows
that, even in weakly anisotropic media, the ray trajec-
tory governed by the anisotropic Snell’s law is signifi-
cantly different from that obtained using the isotropic
form. This could have important implications for such
considerations as imaging (e.g., migration) and lithology
analysis (e.g., amplitude variation with offset).
Expressions are shown specifically for compressional
(qP) waves but they can easily be extended to SH waves
by equating the anisotropic parameters (i.e., ε = δ ⇒ γ )
and to qSV and converted waves by similar means.
The analytic expressions presented are more compli-
cated than the standard form of Snell’s law. To facilitate
practical application, we include our Mathematica code.
INTRODUCTION
The laws governing ray bending have been investigated since
the time of Ptolemy (2nd century A.D.). Prior to the work of
Snell (1591–1626), ray bending had been described by equating
the ratio of angles, and not their sines, to the ratio of veloci-
ties, quite accurate nevertheless for small angles. Snell’s law
extended the validity to all angles of incidence. However, it
assumes both media to be isotropic. In the present paper, we
extend Snell’s expression to encompass anisotropic media.
For simplicity, we assume horizontally layered media and a
dependence of velocity on angle of incidence that is the same
Presented at the 67th Annual International Meeting, Society of Exploration Geophysics. Manuscript received by the Editor March 30, 1998; revised
manuscript received September 27, 1999.
∗
The University of Calgary, Dept. of Mechanical Engineering, Calgary, Alberta T2N 1N4, Canada. E-mail: geomech@telusplanet.net.
‡The University of Calgary, Dept. of Geology and Geophysics, Calgary, Alberta T2N 1N4, Canada. E-mail: slawinski@geo. ucalgary.ca; jbrown@geo.
ucalgary.ca.
∗∗
Formerly Baker Atlas International, Inc., Calgary, Alberta, Canada; presently PanCanadian Petroleum Ltd., 150–9th Ave. SW, Calgary, Alberta,
Canada. E-mail: john parkin@pcp.ca.
c°
2000 Society of Exploration Geophysicists. All rights reserved.
for all azimuths, that is, the media considered exhibit transverse
isotropy (TI) with a vertical symmetry axis (TIV). Although
this case has 2-D geometry, the general methodology is equally
applicable to three dimensions.
We proceed by two different routes: (1) by applying the con-
tinuity of k
x
, the component of the wave number, k, tangential
to the interface between two media, and (2) via Fermat’s princi-
ple, or the stationarity of traveltime between the source located
in one medium and receiver located in the other medium. Both
routes (the continuity conditions and the stationary traveltime
concept) yield the same expression for the refraction law, i.e.,
Snell’s law.
Thus, Snell’s law in horizontally layered media and for a
given frequency can conveniently be restated as the require-
ment that k
x
be continuous across the boundary. This property
holds for both isotropic and anisotropic media regardless of
the type of body wave generated at the boundary (longitudinal
or transverse) and forms the basis for our strategy of calculat-
ing reflected and transmitted angles. The approach following
Fermat’s principle can be formulated through the calculus of
variations (e.g., Slawinski and Webster, 1999).
Present exploration methods of data acquisition and pro-
cessing offer the potential to investigate certain subtle charac-
teristics of the subsurface. Consequently, rigorous and widely
applicable tools are needed. A refraction law for anisotropic
media contributes to more refined and efficient exploration
techniques.
METHOD AND RESULTS
The formulation presented in the next two sections (geomet-
rical and mathematical) states equations for a general velocity
anisotropy expressed in terms of 3-D Cartesian coordinates.
Geometrical formulation
Snell’s law can be illustrated using phase-slowness surfaces
for both incident and transmitted media (e.g., Auld, 1973;
632

Snell’s Law in Anisotropic Media 633
Helbig, 1994). The geometrical construction is facilitated by the
fact that the phase-slowness vectors of the incident, reflected,
and refracted waves are coplanar. Their being coplanar follows
from the kinematic requirement that boundary conditions be
satisfied at all times and at every point of the interface. For
TI therefore, and without any loss of generality, it is possi-
ble to choose a Cartesian coordinate system such that all the
phase-slowness vectors lie in the xz-plane. The familiar case of
isotropic media is considered in Figure 1.
Mathematical formulation
The geometrical approach illustrated in Figure 1 for the
isotropic case (i.e., spherical slowness surfaces) is extended
to include more general scenarios where the slowness sur-
face is an arbitrary surface in slowness space. Consider two
anisotropic media separated by a planar, horizontal interface.
Let the phase-slowness surface in the upper medium be given
by the level surface of a function f (x, y, z) in slowness space
FIG. 1. The geometrical construction yielding reflection and
transmission angles of slowness vectors in an isotropic medium
using the phase-slowness curve. The curves represent phase
slowness in the medium of incidence and transmission, whereas
the Cartesian axes x and z correspond to the horizontal and
vertical slownesses, respectively. The same concept applies in
an anisotropic medium except that the xz-plane cross-section
of the phase-slowness surface does not, in general, form a
sphere. The thin lines within the circles (radii) are collinear
with the phase-slowness vectors; the thick lines, normal to
the phase-slowness surface correspond to the group-slowness
vectors. The symbols ϑ
i
,ϑ
r
and ϑ
t
are the angles between
phase-slowness vectors for incident, reflected, and transmit-
ted waves and the normal to the interface; θ
i
,θ
r
and θ
t
are the
angles between group-slowness vectors for incident, reflected,
and transmitted waves and the normal to the interface (i.e., ray
angles). In the isotropic case, ϑ
i
= θ
i
,ϑ
r
=θ
r
,and ϑ
t
= θ
t
.
spanned by Cartesian coordinates x, y, and z:
f (x, y, z) = a. (1)
Similarly, let the phase-slowness surface in the lower medium
be given by the level surface of a function g(x, y, z) in slowness
space:
g(x, y, z) = b. (2)
In slowness space, x, y, and z have dimensions of slowness, as
does r, which we use below as a general symbol for slowness.
Consider a ray incident on the boundary from above. All
phase-slowness vectors (for incident, reflected, and transmitted
waves) must be coplanar so, without loss of generality, we can
take them to lie in the xz-plane (Figure 2). Denoting the phase-
slowness vector as m, the continuity conditions require that
m
i
·
¯
x = m
r
·
¯
x = m
t
·
¯
x, (3)
where
¯
x is a unit vector in the x-direction and subscripts
i, r, and t refer to incident, reflected, and transmitted waves,
respectively.
The group(ray)-slowness vector, w, is normal to the phase-
slowness surface. Using the property that the gradient points
FIG. 2. The geometrical construction yielding reflection and
transmission angles of slowness vectors in an anisotropic
medium using the phase-slowness curve. The curves represent
phase slowness in the medium of incidence and transmission,
whereas the Cartesian axes x and z correspond to the hor-
izontal and vertical slownesses, respectively. This is an illus-
tration of ray angles for incident, reflected, and transmitted
rays in anisotropic media separated by a horizontal, planar in-
terface using phase-slowness surfaces described by functions
f and g.Themvectors correspond to phase-slowness and
w vectors to group slowness; θ and ϑ correspond to ray an-
gle and phase angle, respectively for incident, reflected, and
transmitted waves. Note that ϑ
i
6= θ
i
,ϑ
r
6= θ
r
, and ϑ
t
6= θ
t
(cf. Figure 1).

634 Slawinski et al.
in the direction in which f increases most rapidly and that it is
normal to any surface of constant f gives
w
i
k∇ f(x,y,z)|
(x
i
,y
i
,z
i
)
, (4)
i.e., the ray vector, w, is parallel to the gradient. Normalizing,
and choosing the function f to have a minimum at the origin
O(0,0,0) and to be monotonically increasing outwards yields
¯
w
i
=−
∇f(x,y,z)|
(x
i
,y
i
,z
i
)
|∇ f (x, y, z)k
(x
i
,y
i
,z
i
)
, (5)
where the negative sign ensures that the incident unit ray vec-
tor,
¯
w
i
, points towards the boundary. From the definition of
dot product, the cosine of the angle of incidence (the angle be-
tween the ray vector and the normal to the interface) is given
by
cos θ
i
= (−
¯
z) ·
¯
w
i
=
¯
z ·∇f(x,y,z)|
(x
i
,y
i
,z
i
)
|∇ f (x, y, z)k
(x
i
,y
i
,z
i
)
, (6)
where
¯
z is a unit vector in the z-direction.
The normalized transmitted ray vector is given by
¯
w
t
=
∇g(x, y, z)|
(x
t
,y
t
,z
t
)
|∇g(x, y, z)k
(x
t
,y
t
,z
t
)
, (7)
and thus the cosine of the angle of transmission (again with
respect to the interface normal) is given by
cos θ
t
= (−
¯
z) ·
¯
w
t
=−
¯
z·∇g(x,y,z)|
(x
t
,y
t
,z
t
)
|∇g(x, y, z)k
(x
t
,y
t
,z
t
)
. (8)
In evaluating the above expression, one uses the facts that
the horizontal phase-slowness components are equal (i.e., x
t
=
−x
i
), and that y
i
and y
t
are zero by the choice of the coordinate
system. Hence, z can be found by substituting x and y into
equations (1) or (2). Note that physical solutions must have
¯
w
t
· (−
¯
z) ≥ 0.
While the phase-slowness vectors, m, are coplanar for the
incident, reflected, and transmitted waves, the ray vectors, w,
need not lie in the same plane. Their directions are deter-
mined by the normals to the phase-slowness surfaces. They will,
however, remain in the same plane if the phase-slowness sur-
faces are rotationally symmetric about the z-axis. If the phase-
slowness surface does not possess rotational symmetry, the in-
cident, reflected and transmitted group vectors need not be
coplanar. In such a case the angle of deviation, χ, from the
sagittal plane, assumed to coincide with the xz-plane and con-
taining all phase-slowness vectors, m, can be found by consid-
ering the projection, w
xz
, of the ray vector, w, on this plane:
w
xz
= [w · x,0,w · z]. (9)
From the definition of scalar (dot) product, it follows that
cos χ =
w · w
xz
|wkw
xz
|
. (10)
The geometrical formulation presented above can be
adapted for incident, reflected, or transmitted rays. Other con-
cepts, such as total internal reflection, also emerge naturally
from this formulation.
QUASI-COMPRESSIONAL WAVE
To illustrate the approach presented above, we derive a
refraction law for quasi-compressional (qP) waves. In equa-
tion (11), qP-wave phase velocity is expressed in terms of two
anisotropic parameters, δ and ε (Thomsen, 1986). In the mathe-
matical formulation, we have expressed angles of incidence and
transmission as a function of the ray parameter, m
x0
or x
0
, com-
mon to both media.
The phase velocity, v
qP
,ofaqP-wave in a weakly anisotropic
medium is given by Thomsen (1986) as
v
qP
(ξ) = α(1 + δ sin
2
ξ cos
2
ξ + ε cos
4
ξ), (11)
where α is the velocity for propagation perpendicular to the
interface, and δ and ε are anisotropic parameters. Here, the
phase angle, ξ, is the phase latitude, the complement of
the phase colatitude, ϑ, used by Thomsen (1986).
The slowness curve, m ≡ r, in the medium of incidence can
be expressed as
r(ξ) =
1
v
qP
(ξ)
=
1
α(1 + δ sin
2
ξ cos
2
ξ + ε cos
4
ξ)
. (12)
In the 2-D TIV case, using a standard expression for the
normal to a curve expressed in polar coordinates, we use an
equation relating ray angle of incidence, θ, and the phase lati-
tude, ξ:
θ = arctan
dr
dξ
− r tan ξ
dr
dξ
tan ξ + r
, (13)
(e.g., Anton, 1984).
Knowing the characteristics of the medium of incidence
(α
i
,δ
i
,ε
i
) (using the subscript i or t to refer to the medium
of incidence or transmission, respectively), and given the ray
angle of incidence, θ
i
, equation (13) still can not, in general, be
solved explicitly for the corresponding phase angle, ξ
i
. It can,
however, be solved numerically (see Appendix A). Once the
phase angle, ξ
i
, is found, the corresponding ray parameter, x
0
,
can be calculated as
x
0
=
cos ξ
i
v
qP
(ξ
i
)
=
cos ξ
i
α
i
¡
1 + δ
i
sin
2
ξ
i
cos
2
ξ
i
+ ε
i
cos
4
ξ
i
¢
.
(14)
Having found the ray parameter, x
0
, which is continuous
across the interface, we calculate the phase latitude in the
medium of transmission, ξ
t
, from
x
0
=
cos ξ
i
v
qP
(ξ
i
)
=
cos ξ
t
v
qP
(ξ
t
)
=
cos ξ
t
α
t
¡
1 + δ
t
sin
2
ξ
t
cos
2
ξ
t
+ ε
t
cos
4
ξ
t
¢
. (15)
Equation (15) can be rewritten as a quartic in cos ξ
t
and solved
for the phase angle, ξ
t
;
α
t
x
0
(ε
t
− δ
t
) cos
4
ξ
t
+ α
t
x
0
δ
t
cos
2
ξ
t
− cos ξ
t
+ α
t
x
0
= 0,
(16)
where physically acceptable values of cos ξ
t
must be real and
in the range [0, 1].
Having found the phase latitude in the medium of transmis-
sion, ξ
t
, and knowing that the transmitted ray is normal to the
slowness curve, we can calculate the ray angle of transmission,

Snell’s Law in Anisotropic Media 635
θ
t
, explicitly from equation (13). The above method was per-
formed based on the weak-anisotropy assumption, given the
properties of the media of incidence and transmission.
We can also proceed using a more general mathematical for-
mulation. TIV media (i.e., 2-D velocity anisotropy) can be char-
acterized by a slowness curve. Consequently, for such cases,
the former approach is sufficient. For potentially more compli-
cated media (i.e., exhibiting a 3-D velocity anisotropy), a more
general method, involving slowness surfaces (as opposed to
slowness curves) is necessary. To illustrate its use based on the
TI-media case, let g(r,ξ) be a function in slowness space de-
fined by
g(r,ξ)=
1
r
−α
t
¡
δ
t
sin
2
ξ cos
2
ξ + ε
t
cos
4
ξ
¢
, (17)
where r is the radius of the slowness surface (i.e., the magni-
tude of the slowness). The slowness curve is given by the level
surface of g(r,ξ), i.e., by the set of points (r,ξ) for which
g(r,ξ)= α
t
. (18)
The gradient can be expressed as
∇g(r,ξ)=r
∂g
∂r
+4
1
r
∂g
∂ξ
, (19)
where the angle is measured with respect to the x-axis, r is
the radial unit vector, and 4 is the azimuthal unit vector (i.e.,
perpendicular to the radius).
The propagation (ray, group) vector is always perpendicular
to the slowness surface (i.e., its direction is parallel to the gra-
dient of g). For g given by expression (17), the gradient can be
written as follows:
∇g(r,ξ)= r
µ
−
1
r
2
¶
+4
Ã
α
t
sin
¡
2ξ
¢£
δ
t
cos(2ξ) − 2ε
t
cos
2
ξ
¤
r
!
.
(20)
In polar coordinates, the Cartesian unit vector, z, can be ex-
pressed as
z = r sin ξ + 4 cos ξ. (21)
This form is used in the desired dot product, i.e., equation (8).
The transmitted group angle, θ
t
, which the group slowness vec-
tor makes with the normal to the interface, can be expressed in
terms of the given transmitted phase angle, ξ
t
, calculated using
equation (16), as
cos θ
t
=
α
t
cos ξ
t
sin(2ξ
t
)
¡
δ
t
cos(2ξ
t
) − 2ε
t
cos
2
ξ
t
¢
−
sin ξ
t
r(ξ
t
)
s
1
£
r(ξ
t
)]
2
+ α
t
sin(2ξ
t
)
£
δ
t
cos(2ξ
t
) − 2ε
t
cos
2
ξ
t
¤
2
,
(22)
where,
r(ξ
t
) =
1
α
t
¡
1 + δ
t
sin
2
ξ
t
cos
2
ξ + ε
t
cos
4
ξ
t
¢
. (23)
Numerical example
Consider a planar horizontal interface between two anisotropic
media. In the upper medium, the vertical wave speed is
α
1
= 3000 m/s, and the anisotropic parameters are ε
1
=−0.2
and δ
1
= 0.1. In the lower medium, the vertical wave speed is
α
2
= 4000 m/s, and anisotropic parameters are ε
2
= 0.15 and
δ
2
=−0.2.The ray strikes the interface from above (Figure 3)
at an incidence angle of θ
i
= 30
◦
. The calculations (Appendix
A) yield the results in Table I.
Notice that based on vertical wave speeds, as would be the
case for α
1
and α
2
obtained from “zero-offset” vertical seismic
profiles (VSP), an incidence angle of 30
◦
would yield (by the
standard form of Snell’s law) a refraction angle of 41.81
◦
. This is
significantly different from the result of 64.01
◦
obtained based
on the weak-anisotropy assumption.
Particularly complicated phenomena can be observed in the
case of converted waves (qP to qSV). For instance, at a flat
horizontal interface, a ray can bend towards or away from the
normal, depending on the angle of incidence (e.g., Slawinski,
1996). In particular cases, such bending leads to nonuniqueness
of raypath, i.e., given a source and receiver, there are several
stationary points of the traveltime function and consequently
several ray trajectories.
There are several ways to confirm the correctness of the solu-
tion (i.e., to verify that the results obtained using an algorithm
Table 1. Group and phase angles and velocities of incidence
and transmission.
Incidence Transmission
Group angles 30.00
◦
64.01
◦
Phase angles 35.57
◦
51.53
◦
Group velocities 3013 m/s 4133 m/s
Phase velocities 2998 m/s 4036 m/s
FIG. 3. Quasi-compressional (qP) wave slowness curves (i.e.,
cross-sections of corresponding slowness surfaces in a ver-
tical plane). Parameters are α
1
= 3000 m/s, α
2
= 4000 m/s,
ε
1
=−0.2,ε
2
=0.15,δ
1
=0.1,δ
2
=−0.2. The outer curve corre-
sponds to the “slower” medium of incidence. The inner curve
corresponds to the “faster” medium of transmission.

636 Slawinski et al.
are in agreement with certain fundamental requirements). The
fulfillment of those requirements constitutes necessary condi-
tions for the validity of the method. First, the phase and group
angles, ϑ, and θ, as well as the magnitudes of phase and group
velocities, v and V, must satisfy the following equation in either
medium:
cos(θ − ϑ) =
¯
¯
¯
¯
v(ϑ)
V(θ )
¯
¯
¯
¯
. (24)
This can be confirmed graphically with the help of Figure 4,
from which it is clear that
cos ζ = v/V. (25)
Second, the phase angles and phase velocities must satisfy the
following equation across the interface:
sin(ϑ
1
)
v
1
(ϑ
1
)
=
sin(ϑ
2
)
v
2
(ϑ
2
)
, (26)
where the subscripts 1 and 2 correspond to the upper and lower
media, respectively. Expression (26) is the standard form of
Snell’s law, which is always valid for phase angles and phase
velocities. Third, Fermat’s principle of stationary time must be
satisfied (e.g., Helbig, 1994). This might not be obvious from a
quick inspection, and it might require a reformulation in terms
of the Euler-Lagrange equation (e.g., Slawinski and Webster,
1999).
The approach involving exact formulas
In principle, it is possible to carry out all the derivations de-
scribed in this paper using exact equations. For qP-wave phase
velocity, the form equivalent to equation (11) is (Thomsen,
1986)
F(r,ξ) =
1
r
−α
q
1+εcos
2
ξ + D(ξ) = 0, (27)
where D(ξ) is given by
D(ξ ) ≡
1
2
µ
1 −
β
2
α
2
¶
×











v
u
u
u
u
u
u
t
1 +
4(2δ − ε) sin
2
ξ cos
2
ξ
1 −
β
2
α
2
+
4ε
µ
1 −
β
2
α
2
+ ε
¶
cos
4
ξ
µ
1 −
β
2
α
2
¶
2
− 1











. (28)
DISCUSSION AND CONCLUSIONS
In this paper, we have illustrated the derivation and applica-
tion of the general Snell’s law using compressional (qP) waves.
The same approach, however, can be applied in an analogous
manner to both shear and converted waves (Slawinski, 1996).
For SH-waves, whose velocity dependence can be described
in terms of a single parameter, γ , one can quite conveniently
use an exact expression without the weak-anisotropy approxi-
mation. Notably, the exact expression for a refraction law with
elliptical velocity dependence (equivalent to SH-waves in TI
media) was presented as a result of a VSP study in the Sahara
desert (Dunoyer de Segonzac and Laherrere, 1959). Also, one
FIG. 4. The phase-velocity vector, v, is orthogonal to the wave-
front. The group-velocity vector, V, in an anisotropic yet ho-
mogeneous medium is collinear with the ray under considera-
tion. For an infinitesimal time increment, greatly exaggerated
here (l
0
to l
1
), V is the hypotenuse of a right triangle, and the
enclosed angle ζ is equal to θ − ϑ .
can easily reduce expressions provided in this paper to the
weak-anisotropy SH case by equating parameters ε and δ.
ACKNOWLEDGMENTS
The authors acknowledge the considerable support of
the University of Calgary and the CREWES Project. Also,
the authors acknowledge the critical sponsorship of The
Geomechanics Project, namely, Baker Atlas, Integra Scott
Pickford, PanCanadian Petroleum, Petro-Canada Oil and Gas,
and Talisman Energy. In addition, we thank Don Lawton and
an anonymous reviewer for their critical review and cogent
suggestions, and Larry Lines for his stewardship of the review
process.
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Snell’s Law in Anisotropic Media 637
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Wolfram, S., 1991, Mathematica, a system for doing mathematics by
computer, 2nd edition: Addison-Wesley Publishing Co.
APPENDIX A
MATHEMATICA CODE FOR qP WAVES
Mathematica software can be used to investigate ray bend-
ing at the interface between two (TI) anisotropic media. Given
angle of incidence and parameters of the medium of inci-
dence and the medium of transmission, it provides the angle
of transmission, s. Note that Mathematica software is case-
sensitive.
Qi = angle of incidence in radians
VJ = vertical speed in the medium of incidence
VD = vertical speed in the medium of transmission
EJ = anisotropic parameter in the medium of incidence, ε
DJ = anisotropic parameter in the medium of incidence, ε
ED = anisotropic parameter in the medium of transmission, δ
DD = anisotropic parameter in the medium of transmission, δ
R = Qi (initial guess for the numerical solution)
the next step expresses the slowness curve in the medium of in-
cidence:
ri = 1/(VJ*(1+DJ*Sin[zi]
∧
2*Cos[zi]
∧
2+EJ*Cos[zi]
∧
4))
the next step calculates the derivative w.r.t. phase angle in slow-
ness curve in the medium of incidence:
dri = Sin[2*zi]*(EJ−DJ*Cos[2*zi]+EJ*Cos[2*zi])/
(VJ*(1+EJ*Cos[zi]
∧
4+DJ*Cos[zi]
∧
2*Sin[zi]
∧
2)
∧
2)
next steps calculate the ray parameter, x:
FindRoot[Cot[Qi]==
(dri−ri*Tan[zi])/(dri*Tan[zi]+ri), {zi,R,−Pi,Pi}]
zif = Abs[zi/.%]
x = N[Abs[Cos[zif]/
(VJ*(1+DJ*Sin[zif]
∧
2*Cos[zif]
∧
2+EJ*Cos[zif]
∧
4))]]
the next step calculates the phase latitude in the medium of trans-
mission, z:
FindRoot[VD*x*(ED−DD)*C
∧
4+VD*x*DD*C
∧
2−
C+VD*x == 0, {C, 0.5, 0, 1.5}]
z = ArcCos[C/.%]
dz = 2*z
r = 1/(VD*(1+DD*Sin[z]
∧
2*Cos[z]
∧
2+ED*Cos[z]
∧
4))
t = VD*Cos[z]*Sin[dz]*(Abs[DD*Cos[dz]−2*ED*Cos[z]
∧
2])
−Sin[z]/r
b = Sqrt[1/r
∧
2+(VD*Sin[dz]*(DD*Cos[dz]
−2*ED*Cos[z]
∧
2))
∧
2]
the next step calculates the angle of transmission,s
s=ArcCos[Abs[t/b]]