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February 15, 2012 / Vol. 37, No. 4 / OPTICS LETTERS
701
Imaging with complex ray-optical
refractive-index interfaces
between complex object and image distances
Johannes Courtial,1,* Blair C. Kirkpatrick,1 and Miguel A. Alonso2
1
School of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, United Kingdom
2
The Institute of Optics, University of Rochester, Rochester, New York 14627, USA
*Corresponding author: [email protected]
Received December 2, 2011; revised January 6, 2012; accepted January 9, 2012;
posted January 10, 2012 (Doc. ID 159258); published February 14, 2012
Building on recent work on windows that can be modeled as interfaces between materials with a complex rayoptical refractive-index ratio, we define here objects and images at complex positions. We give an example of
an optical component that performs imaging between real and complex object and image positions. © 2012 Optical
Society of America
OCIS codes: 080.0080, 080.2720, 110.2990.
Imagine a transparent window that does not offset light
rays, but that rotates them by an angle α 90° around
the local window normal. A wave-optical consideration
[1] reveals that such ray-rotating windows, if they existed, could create wave-optically impossible light-ray
fields: light-ray direction is given by the gradient of the
eikonal, ∇ϕ; for the example of an incident cylindrical
wave front with a quadratic phase dependence of the
form ϕx; y ∝ x2 ∕2 in the plane of the window, the field
of light-ray directions there is given by px; y ∝ x; 0; pz (we do not have enough information to calculate the z
component, but this is not necessary). After transmission
through the window, the light-ray-direction field is
p0 x; y ∝ 0; x; pz . The z component of the curl of this
light-ray field is ∇ × p0 x; y ∝ ∂∕∂xx − ∂∕∂y0 1,
but as ∇ × ∇ϕ0 0 for all functions ϕ0 almost everywhere, no eikonal ϕ0 x; y can be constructed for the
resulting light-ray field such that p0 ∇ϕ0 .
An offset in the ray position that accompanies the lightray-direction change can overcome this problem. For example, it is clearly possible to rotate the light-ray field in
the above example as a whole. It is also possible to make
the ray-position offset small, for example by rotating the
incident wave front in small pieces. This approach of piecewise wave-front transformation is taken by so-called
“metamaterials for rays” (METATOYs) [1] (but note that
METATOYs are not metamaterials in the standard sense
[2]). If each of these pieces is much larger than the wavelength but much smaller than the scale or variations for
the incident ray-direction field, they can be as unnoticeable as a good monitor’s pixels. In this way, METATOYs
can mimic wave-optically forbidden windows.
A number of METATOYs performing various forbidden light-ray-direction changes have been investigated
theoretically [3–10]. So far, METATOYs performing
three different light-ray-direction changes have been
demonstrated experimentally: namely, flipping of one
transverse ray-direction component [11], ray-optical refraction with a variation of Snell’s law in which the sines
are replaced by tangents [12], and ray rotation around the
0146-9592/12/040701-03$15.00/0
window normal (through 90°, as already described
above, and through other, in principle arbitrary,
angles) [13].
Ray rotation around the local normal is particularly
interesting, as it can be described formally as refraction
at an interface between materials with a complex rayoptical refractive-index ratio [14]. Note that we call the
refractive-index ratio ray-optical, as its effect is quite
different from that of a material with a complex (standard) refractive index, which is associated with absorption [15]. The experimental demonstration of windows
Fig. 1. (Color online) Transformation of a ray cone originating
from a point light source at P into a ray hyperboloid by passage
through a ray-rotating window, for different ray-rotation angles,
α. Light-ray trajectories are shown as cylinders. After transmission through the window, the rays lie on the surface of hyperboloids. For α 90°, the hyperboloid’s waist plane coincides
with the window plane. For α 180°, the hyperboloid is degenerate: a cone. The images were created using the ray-tracing
software TIM (The Interactive METATOY) [16].
© 2012 Optical Society of America
702
OPTICS LETTERS / Vol. 37, No. 4 / February 15, 2012
representing a complex ray-optical refractive-index interface [13] has prompted us to study the consequences of
such unusual refraction for ray optics.
Here we describe an extension of geometrical imaging
to complex object and image distances. We confirm that
our description works for imaging between real object
and image distances using parallel ray-rotation windows
[6]. We believe that this work has the potential to lead to
new optical design paradigms.
We are considering a planar window that rotates light
rays by an angle α around the window normal. Specifically, we are interested in its effect on rays from a point
light source. A point light source emits light rays in all
directions; any such light ray lies on a cone with its apex
at the light-source position and its axis perpendicular to
the ray-rotating window. Figure 1 shows ray-tracing simulations of light rays on one such cone before and after
transmission through such windows. In the case α 180°, the rays behind the window form another cone with
its apex at the image of the point light source. For all
other nontrivial cases, the rays lie on the surface of a hyperboloid (of one sheet). (Trivial cases include α 0°
and the case of a point light source in the plane of the
window, and in all of these, the rays remain on the original cone.)
Now consider a more general case, in which the rayrotating window converts one ray hyperboloid into another ray hyperboloid (Fig. 2(a)). Figure 2(b) shows
the trajectory of one light ray for this case, orthographically projected into the window plane. The projection of
other rays on the same hyperboloids can be obtained by
rotating the projection of the shown trajectory around
the point A, which is the projection of the common axis
of both hyperboloids. The incident light ray intersects the
window at the point I, where it gets rotated through an
angle α around the window normal. As the (local) window normal is the axis around which the light ray is rotated, the angle θ between the light ray and the window
normal does not change in this process.
We define the point P as the center of the hyperboloid
in front of the window. As it lies on the hyperboloid’s
Fig. 2. (Color online) Transformation of one ray hyperboloid
into another by transmission through a window that rotates ray
direction through an angle α around the window normal. (a) 3D
view of 30 rays (cylinders). (b) Orthographic projection of a representative ray (thick arrows) into the window plane. A, P, P 0 ,
and I are, respectively, the projections of the axis common to
both hyperboloids, of the centers of the two hyperboloids, and
of the point where the ray intersects the window. zr is the distance of P in front of the window, and z0r is the distance of P 0
behind the window. θ is the angle between the incident light ray
and the window normal, which is also the angle between the
outgoing light ray and the window normal (as the ray-rotation
axis is a window normal). d and d0 are the skirt radii of the two
hyperboloids.
axis, its projection is the same as that of the axis. Similarly, we define the point P 0 as the center of the hyperboloid behind the window. We call the distance of P
in front of the window zr (if P is in front of the window,
zr > 0, otherwise zr ≤ 0) and that of P 0 behind the window z0r (z0r > 0 when P 0 is behind the window). The
distance of the rays from P and P 0 (and also of the projections of the rays from the projections of P and P 0 ) is,
respectively, d and d0 .
We calculate the dependence of z0r and d0 on zr , d, θ,
and α as follows. We place a complex coordinate system
into the projection plane with its origin at the projection
of I and its real axis pointing in the opposite direction of
the projection of the incident ray. Following [9], we can
then represent the projection of P and P 0 by the complex
numbers
Z P zr tan θ − id;
Z P 0 − expiαz0r tan θ id0 :
(1)
As the projections of P and of P 0 are the same, Z P Z P0 .
After division by tan θ, this becomes
d
d0
0
:
− expiα zr i
zr − i
tan θ
tan θ
(2)
Now we define the complex object and image
distances
z zr i
d
;
tan θ
z0 z0r i
d0
:
tan θ
(3)
A positive imaginary part of a complex object or image
distance implies rays that twist counterclockwise; a negative imaginary part implies clockwise twisting. Eq. (2)
can then be written as
z0 − exp−iαz :
(4)
This equation, which describes imaging between “point”
light sources at complex positions, is the first key result
of this paper.
Note that Eq (4) does not contain the angle θ between
the light rays and the window normal. This means that it
holds not only for light rays on one particular ray hyperboloid, but for all light rays that correspond to the point
light sources at the complex object and image distances
the window images into each other.
Which are those light rays that correspond to a particular point light source at a complex distance with
imaginary part zi ? It can be seen from Eq. (3) that
zi d∕ tan θ. In the form d zi tan θ, this allows calculation of the distance d from the hyperboloid center at
which light rays that travel at an angle θ with respect
to the z direction travel. Special cases include light rays
that travel in the z direction (θ 0°), which pass through
the hyperboloid center (d 0), and light rays that travel
at an angle 45° with respect to the z direction, which have
a distance d zi from the hyperboloid center.
We now test our complex imaging equation, Eq. (4), in
a number of special cases, each involving a point light
February 15, 2012 / Vol. 37, No. 4 / OPTICS LETTERS
source at a real object distance z in front of a ray-rotating
window. In the trivial case of a window that does not actually change the light-ray direction (α 0°), the image
distance correctly comes out as z0 −z. In the case
α 90°, z0 iz, which means that the ray hyperboloid’s
waist plane is in the window plane, which is confirmed
visually by Fig. 1(b). Finally, in the case α 180°
(Fig. 1(c)), which describes the light-ray-direction
change in negative refraction [4,9], the image should
be at the same distance as the object but on the other
side of the window, and indeed z0 z.
We also test whether our formulation correctly describes imaging with two parallel ray-rotating windows
(ray-rotation angles α and β, separation s). Only point
light sources at a real distance
z
−s sin β
sinα β
5
in front of the first window are imaged to a real distance
by such a combination of ray-rotating windows [6].
According to our formalism, the first window images
the point light source to a complex distance
z01
s sin β
:
exp−iα
sinα β
(6)
This intermediate image then acts as a point light source
at a distance
z2 s − z0
1 exp−iβ
s sin α
sinα β
(7)
in front of the second window. Note that the complex
conjugate of z01 is used such that the imaginary part of
the object distance z2 is the same as that of the image
distance z01 , which means that the handedness of the
ray hyperboloid is unchanged. The intermediate object
then gets imaged to a real distance
z0 −s sin α
sinα β
8
behind the second window, which agrees with the result
in [6].
Finally, we use the fact that a ray-rotating window can
be described as the interface between complex refractive
indices n1 and n2 , where [14]
expiα n1
:
n2
(9)
703
We can therefore write Eq. (4) in the form
z0 −
n2 z :
n1
(10)
Eq. (10) is the second key result of this paper. What
makes this form significant is that this equation is also
true if the refractive-index ratio is not in the form of
Eq. (9), but only paraxially. It is therefore a generalization of the standard equation describing paraxial imaging
by a planar refractive-index interface, which differs from
Eqn. (10) only in the absence of the complex conjugation
of z. Remarkably, this generalization holds exactly in the
nontrivial case of ray-rotating windows.
In summary, we have extended the laws of imaging at
planar refractive-index interfaces to complex values of
the refractive-index ratio and of object and image distance. Experimental tests should be possible using rayrotating windows, which were recently demonstrated
experimentally [13].
Thanks to Greg Forbes (QED Technologies) for valuable discussions.
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