COMSOL 6.4 - Alvarez Lens

Freeform optics has generated renewed interest due to advances in fabrication and manufacturing technology. This model studies a specific freeform design called the Alvarez lens, where two complementary cubic surfaces are shifted laterally with respect to each other to achieve variable optical power. Departure from conventional (mostly spherical) optical surfaces provides additional degrees of freedom, can open up many opportunities for systems design, and achieve better optical performance. For example, in Ref. 1 a modified Alvarez lens is used to design a miniature multimodal microscope with variable depth focusing in a small form factor.

Model Definition

The model uses a design described in Ref. 2, in which the surface sags are described as

where t1(X, Y) and t2(X, Y) are the surface sags of the first and the second elements; a is a surface coefficient; X and Y are the normalized coordinates

and rn is the normalizing radius. For transverse shifts of ΔX, ΔY and zero longitudinal shift, the total contribution to phase for both surfaces is given by

Note that for ΔX = 0, the total profile becomes quadratic and proportional to ΔY:

A longitudinal shift of zero is not physical and therefore the optical power and the aberrations introduced in the model deviate slightly from the expressions above. Geometry module can be used to easily parameterize and model these effects. The parameters shown in Table 1 are used for the initial values and a parametric sweep varies the relative shift in the y direction to observe the change in the optical power. The geometry shown in Figure 1 is generated using the parameters in Table 1.

Table 1: parameter definitions.
Parameter	Value	Description
rn	20 mm	Normalizing radius
thickness	5 mm	Central thickness
a	0.43 mm	Surface coefficient
n	1.505	Refractive index
Δx	0	Relative shift, x-coordinate
Δy	-4 mm	Relative shift, y-coordinate
Δz	2 mm	Relative shift, z-coordinate

Figure 1: Alvarez lens where the cubic surfaces are laterally shifted to achieve variable optical power.

In this model, collimated rays are released with a hexapolar distribution and a parametric sweep is performed to observe the change in the focal point as a function of the lateral shift. Spot diagrams are generated to study the aberrations introduced.

Results and Discussion

Figure 2 shows that a lateral shift in y causes a change in optical power. The corresponding spot diagrams shown in Figure 3 indicate that the aberrations change at every configuration. Ref. 1 discusses strategies for reducing aberrations for the entire range using two Alvarez lens pairs.

Figure 2: Ray trajectories for different lateral shifts demonstrate variable optical power.

Figure 3: Spot diagrams for different lateral shifts, showing that performance varies between configurations.

References

1. S.S. Rege and others, “Application of the Alvarez-Humphrey Concept to the Design of a Miniaturized Scanning Microscope,” Optics Express, vol. 12, no. 12, pp. 2574–2588, 2004.

2. D. Gonzalez-Utrera and others, “Modeling, Fabrication, and Metrology of 3D Printed Alvarez Lenses Prototypes,” Optics Express, vol. 32, no. 3, pp. 3512–3527, 2024.

Ray_Optics_Module/Lenses_Cameras_and_Telescopes/alvarez_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

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
rn	20[mm]	0.02 m	Normalizing radius
thickness	5[mm]	0.005 m	Central thickness
a	0.43[mm]	4.3E-4 m	Surface coefficient
nref	1.505	1.505	Refractive index
dx	0[mm]	0 m	Displacement in x
dy	-4[mm]	-0.004 m	Displacement in y
dz	2[mm]	0.002 m	Displacement in z