Luneburg Lens

In this tutorial the Geometrical Optics interface is used to compute ray trajectories through a graded-index medium. The Luneburg lens has a graded refractive index which leads to special focusing properties.

A Luneburg lens, in the most general sense, is a spherically symmetric thick lens with a variable-index refracting structure that forms perfect geometrical images of two concentric spheres onto each other. The Luneburg lens has a gradient of isotropic refractive index n radially out from its center. In the generalized Luneburg lens, there is a pair of conjugate foci outside the lens.

In the limiting case where one of the foci tends to infinity and the other one is located on the lens surface, the analytical solution for the index profile takes a very simple form. This is what is usually meant by “Luneburg lens” in the narrow sense. Such a lens focuses a parallel beam to a perfect point in the geometrical optics limit. The location of the focus is on the rim of the lens directly opposite to the incidence direction.

Unlike a conventional, constant-index lens, a Luneburg lens works perfectly for wide ray bundles, and not only for paraxial beams. Thus, the lens is said to have a 180 degree field of view and zero f-number (Ref. 1). These properties of a lens can only be achieved using gradient-index optics.

Luneburg lenses can be made from transparent dielectric media for essentially any wavelength of interest, and they can be extremely broadband (Ref. 1). At microwave frequencies, they are used as small form-factor focusing devices for high-fidelity satellite antennas. Unlike a parabolic reflector dish, a Luneburg lens can focus satellite signal arriving from any position in the sky, which makes it more suitable for satellite antennas mounted on moving objects, such as trains and ships.

Model Definition

The refractive index of a general Luneburg lens takes the form:

where r is the radial coordinate from the center of the lens and R is the radius of the lens. The dimensionless parameter f determines whether rays are focused inside or outside the lens. For f = 1 the focal point lies on the surface of the lens.

Results and Discussion

A collimated array of rays is released into the sphere.The focal parameter can be adjusted to change the refractive index profile within the lens. For this simulation, the parameter is set to f = 1.

The Luneburg Lens is shown in Figure 1 together with the ray trajectories. The curved path of the rays within the lens can be clearly seen. In this figure the rays have been colored according to their radial location at the best focus position.

The image quality at the focal point can be seen in Figure 2. As expected, with f = 1, the best focus is exactly on the surface of the sphere opposite the direction of the incoming rays.

Figure 1: Ray trajectories in a Luneburg lens. The color indicates optical path length.

Figure 2: A spot diagram for the Luneburg lens. The image quality is essentially perfect.

References

1. N. Kundtz and D.R. Smith, Extreme-Angle Broadband Metamaterial Lens, Nature Materials Letters, 2009.

2. L.D. Landau and E.M. Lifshitz, The Classical Theory of Fields, 4th ed., Butterworth-Heinemann, Oxford, 1975.

Ray_Optics_Module/Lenses_Cameras_and_Telescopes/luneburg_lens

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

In the tree, select .

Click

Global Definitions

Add some parameters for the geometry dimensions.

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
L	1[m]	1 m	Box length
R	0.4[m]	0.4 m	Outer radius

Geometry 1

The Luneburg lens is simply a sphere containing a graded-index medium.

Sphere 1 (sph1)

In the toolbar, click

In the window for , locate the section.

In the text field, type R.

Click

Click the

button in the toolbar.

Click the

button in the toolbar.

Component 1 (comp1)

In the window, click .

In the window for , locate the section.

Find the subsection. From the list, choose . The ray tracing algorithm used by the Geometrical Optics interface computes the refracted ray direction based on a discretized geometry via the underlying finite element mesh. A cubic geometry shape order usually introduces less discretization error compared to the default, which uses linear and quadratic polynomials.

Definitions

Add some expressions for the radius from the center of the lens. This will be used to define the refractive index later.

Variables 1

In the window, under right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Unit	Description
r	sqrt(x^2+y^2+z^2+eps)	m	Radial coordinate
f	1.0		Focal shift parameter
n	sqrt(1+f^2-(r/R)^2)/f		Refractive index

Materials

Material 1 (mat1)

In the window, under right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Refractive index, real part	n_iso ; nii = n_iso, nij = 0	n	1	Refractive index

Geometrical Optics (gop)

In the window, under click .

In the window for , locate the section.

In the text field, type 0. This model is only concerned with the transmitted (refracted) rays and not the reflected ones, so set the maximum number of secondary rays to 0.

Select the check box. In this simulation, the geometry normals are used to apply the boundary conditions on all refracting surfaces. This is appropriate for the highest accuracy ray traces in single-physics simulations, where the geometry is not deformed.

Material Discontinuity 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Release the rays in the x direction in collimated hexapolar grid.

Release from Grid 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Specify the qc vector as


-0.75	x

In the Rc text field, type 0.35.

In the Nc text field, type 10.

Locate the section. Specify the L0 vector as


1	x
0	y
0	z

The default Physics-controlled mesh can be used for this simulation.

Mesh 1

In the window, under right-click and choose .

Study 1

Step 1: Ray Tracing

In the window, under click .

In the window for , locate the section.

From the list, choose .

In the text field, type range(0,0.05,1.75).

In the following steps, adjust the solver settings to ensure that sufficiently small time steps are taken through the graded index medium.

Solution 1 (sol1)

In the toolbar, click

In the window, expand the node, then click .

In the window for , click to expand the section.

From the list, choose .

In the text field, type 2.5e-12. This gives around 1000 time steps within the lens.

Click to expand the section. In the toolbar, click

Results

Ray Diagram 1

In the window for , type Ray Diagram 1 in the text field.

Locate the section. Select the check box.

In the window, expand the node.

Color Expression 1

In the window, expand the node, then click .

In the window for , locate the section.

In the text field, type at(4.5975768e-9,gop.rrel).

From the list, choose .

Surface 1

In the window, right-click and choose .

In the window for , locate the section.

From the list, choose .

Transparency 1

Right-click and choose .

Slice 1

In the window, right-click and choose .

In the window for , locate the section.

In the text field, type n.

Locate the section. From the list, choose .

In the text field, type 1.

Locate the section. From the list, choose .

Clear the check box.

Transparency 1

Right-click and choose .

Slice 2

In the window, under right-click and choose .

In the window for , locate the section.

From the list, choose .

In the text field, type 1.

In the toolbar, click

Click the

button in the toolbar. Compare the resulting plot to Figure 1.

Spot Diagram

In the toolbar, click

and choose .

In the window for , type Spot Diagram in the text field.

Locate the section. Select the check box.

Select the check box.

Click to expand the section. Select the check box.

In the text field, type 8.

Spot Diagram 1

In the toolbar, click

and choose .

In the window for , click to expand the section.

From the list, choose .

In the text field, type 1.

In the text field, type 0.

Click .

Click to expand the section. Select the check box.

From the list, choose .

In the text field, type 6.

In the text field, type 4.

Color Expression 1

Right-click and choose .

In the window for , locate the section.

In the text field, type t. This shows the time at which the RMS focus is minimized. This time was used to define the color expression for the Ray Diagram shown in Figure 1.

In the toolbar, click

Click the

button in the toolbar. Compare the resulting plot to Figure 2.