4.5 Achievement of Phase Matching
97
than o-ray everywhere except along the optic axis, along which v e = v o . For better
understanding and visualization, we have shown index ellipsoids of a negative and
positive uniaxial crystal. An index ellipsoid is a 3D surface plot of refractive index
along all the directions in Cartesian space. Figure 4.7a–b shows the index ellipsoids
of a negative uniaxial crystal for both e-ray and o-ray, respectively. Here the optic axis
has been assumed to be oriented along the z-axis, as per the convention. For e-ray,
refractive index n e < n o along x- and y-directions and n e = n o along the z-direction.
In the figure, the index ellipsoid of the negative crystal is blue in color and of prolate
shape (like a rugby ball). The o-ray index ellipsoid is spherical indicating isotropic
nature, since its refractive index is the same (n o ) in all directions. Figure 4.7c–d.
presents the cross-sectional view of two index ellipsoids put together. It can be
observed that index ellipsoid of the e-ray always remains inside that of the o-ray.
Similarly, the index ellipsoids of a positive uniaxial crystal have been drawn and
shown in Fig. 4.8. In this case, the index ellipsoid of e-ray is oblate in shape (like a
pumpkin) and that of o-ray is again spherical. In a positive crystal, refractive index
of e-ray is equal to n o along the optic axis (the z-axis) and is equal to n e (>n o ) along
x- and y-directions. Hence, in this case, it is the isotropic sphere of o-ray that always
lies inside the ellipsoid of e-ray.
We now know refractive indices of both the types along the three axes, but cannot determine the refractive index of an extraordinary ray traveling in an arbitrary
direction. In that endeavor, let us first think of an ellipse, as shown in Fig. 4.9. From
our knowledge of geometry, we know that the equation of the ellipse is [199]
x
2
a 2 +
y
2
b 2 = 1
(4.73)
where a and b are the semi-major and semi-minor axes of the ellipse and (x, y) are
the coordinates of an arbitrary point P on it. If the length of the line segment OP is
r , then x = rcosθ and y = rsinθ, where θ is the angle OP makes with respect to the
x-axis. Then, Eq. 4.73 can be written as
r
2 cos
2
θ
a 2
+
r
2 sin
2
θ
b 2
= 1
(4.74)
or
1
r 2 =
cos
2
θ
a 2 +
sin
2
θ
b 2
(4.75)
Since r changes w.r.t. θ, we can write r as a function of θ as
1
r 2 (θ)
=
cos
2
θ
a 2 +
sin
2
θ
b 2
(4.76)
Let us analogously extend this equation to the case of index ellipsoids, following
which the refractive index of the e-ray in an arbitrary direction is given by
97
than o-ray everywhere except along the optic axis, along which v e = v o . For better
understanding and visualization, we have shown index ellipsoids of a negative and
positive uniaxial crystal. An index ellipsoid is a 3D surface plot of refractive index
along all the directions in Cartesian space. Figure 4.7a–b shows the index ellipsoids
of a negative uniaxial crystal for both e-ray and o-ray, respectively. Here the optic axis
has been assumed to be oriented along the z-axis, as per the convention. For e-ray,
refractive index n e < n o along x- and y-directions and n e = n o along the z-direction.
In the figure, the index ellipsoid of the negative crystal is blue in color and of prolate
shape (like a rugby ball). The o-ray index ellipsoid is spherical indicating isotropic
nature, since its refractive index is the same (n o ) in all directions. Figure 4.7c–d.
presents the cross-sectional view of two index ellipsoids put together. It can be
observed that index ellipsoid of the e-ray always remains inside that of the o-ray.
Similarly, the index ellipsoids of a positive uniaxial crystal have been drawn and
shown in Fig. 4.8. In this case, the index ellipsoid of e-ray is oblate in shape (like a
pumpkin) and that of o-ray is again spherical. In a positive crystal, refractive index
of e-ray is equal to n o along the optic axis (the z-axis) and is equal to n e (>n o ) along
x- and y-directions. Hence, in this case, it is the isotropic sphere of o-ray that always
lies inside the ellipsoid of e-ray.
We now know refractive indices of both the types along the three axes, but cannot determine the refractive index of an extraordinary ray traveling in an arbitrary
direction. In that endeavor, let us first think of an ellipse, as shown in Fig. 4.9. From
our knowledge of geometry, we know that the equation of the ellipse is [199]
x
2
a 2 +
y
2
b 2 = 1
(4.73)
where a and b are the semi-major and semi-minor axes of the ellipse and (x, y) are
the coordinates of an arbitrary point P on it. If the length of the line segment OP is
r , then x = rcosθ and y = rsinθ, where θ is the angle OP makes with respect to the
x-axis. Then, Eq. 4.73 can be written as
r
2 cos
2
θ
a 2
+
r
2 sin
2
θ
b 2
= 1
(4.74)
or
1
r 2 =
cos
2
θ
a 2 +
sin
2
θ
b 2
(4.75)
Since r changes w.r.t. θ, we can write r as a function of θ as
1
r 2 (θ)
=
cos
2
θ
a 2 +
sin
2
θ
b 2
(4.76)
Let us analogously extend this equation to the case of index ellipsoids, following
which the refractive index of the e-ray in an arbitrary direction is given by