This article shows how the OPTOTUNE focus-tunable liquid lenses are modelled in OpticStudio. It will further demonstrate in an example design how an optical system with a liquid lens can be optimized. In a final section we will show how to model typical optical aberrations into a liquid lens model for a sensitivity check.

Authored By Michael Büeler

The OPTOTUNE lens catalog

The OPTOTUNE lens catalog includes models of five different lens types that vary in focal power range and clear aperture diameter. For each lens type, three different models are provided to represent three specific focus states of the lens: MAX represents the maximum focal power state (shortest positive focal length), MIN represents the minimum focal power state (shortest negative focal length), and MID represents the zero focal power state.

The variable focal power states of focus-tunable liquid lenses will normally be handled with the Multiconfiguration editor of OpticStudio. Since the lens catalog feature does not support multi-configuration models, the above approach with the MAX, MIN and MID models was chosen to initially represent the focal power range of each liquid lens type. In a later step we will show how to easily convert one of the loaded single-state models into a multi-configuration model.

How OPTOTUNE liquid lenses are modelled in OpticStudio

Every liquid lens model comprises the following surfaces:

• Housing: First and last surface of each model. Representing the mechanical envelope of the liquid lens both in axial length and diameter. Can be used as indicators to keep a sufficient distance to the preceding and following elements in the system.
• Cover glass: Protects the variable membrane surface from contamination.
• Variable surface: Surface the radius of curvature of which can be changed within the boundaries noted in the comment line. In reality, this curvature change is achieved by changing the drive current to the lens.
  • Optical liquid: The variable surface is followed by a proprietary optical liquid of a specific refractive index and Abbe number
• Container glass: Contains the optical liquid against the variable surface. It is in contact with liquid on one side, and with air on the other side.

More notes on the liquid lens models:

• Every liquid lens model ranges from surface "Housing" to surface "Housing"
• If required the models can be reversed with the regular reverse function
  
• The surface named "Variable" represents the surface with variable radius of curvature.
  • The minimum and maximum radii of curvature are noted in the Comment cell.
  • Use the Multi-Configuration Editor to define multiple states of the variable surface.
    
  • The optical liquids are taken from the OPTOTUNE.AGF material catalog.
    
  • The material models consider dispersion and refractive index change with temperature.
  • Thermal expansion is ignored since the expansion does not only concern the optically relevant clear aperture but also the hidden periphery of the lens and simulation would be complex. For closed loop applications with auto-focus this is no problem since any temperature-induced focus drift would automatically be compensated. For open loop applications, Optotune is offering a number of thermally calibrated (TC) lenses that are automatically compensating temperature-induced focus drifts by means of an onboard temperature sensor and calibration data on an EEPROM.

The OPTOTUNE liquid lens models are using edge-thickness solves to accurately model the physical movement of the variable surface and the center thickness changes. The edge thickness solves are placed on surfaces 3 and 4 of the models.

The center thicknesses of surfaces 3 and 4 change when the radius of curvature of the variable surface is changed. Through the use of edge-thickness solves these changes are automatically taken care of and the center thickness values do not need to be varied in the Multiconfiguration editor.

In the lenses of the EL-16-40 family, the variable surface is fixed on the outside (edges) while the optical liquid is pumped into and out of the optical area from the lens periphery. For these lenses the edge thickness solve is set at a radial height which corresponds to the semi-diameter of the variable surface.

In the lens types EL-12-30 and EL-3-10, the entire liquid container is moving and both the vertex and the edge of the variable surface are moving. These lens types are most accurately modelled if the edge thickness solve is set at a radial height which corresponds to roughly 2/3 of the semi-diameter of the variable surface.

How to optimize an optical system with an OPTOTUNE liquid lens

The following steps show how to combine a liquid lens model with an optical system (single paraxial lens), how to set up the Multiconfiguration editor and the Merit Function editors, and how to optimize the curvatures of the liquid lens variable surface for best focus at 3 different configurations.

1. Load model of EL-16-40-20D from lens catalog
2. In the Lens Data Editor: Add a paraxial lens with f=60mm and a thickness of 130mm behind the liquid lens. This lens shall represent an existing optical system.
3. Open the Multi-Configuration Editor as shown in the screenshot below and add 2 more configurations
   
4. In the Multi-Configuration Editor: Add the curvature operand CRVT for the variable surface 4 (see screenshot below). E.g. set the start values to the curvatures corresponding to the MIN, MID and MAX radii of curvature values in the Comment cell of the variable surface in the Lens Data
   
5. In the Multi-Configuration Editor: Add the operand THIC for surface 0 (see screenshot below). Set object distances to e.g. 45, 100, and inf
6. Open the Merit Function Editor.
7. In the Merit Function editor: Limit the curvature of the liquid lens variable surface with operands CVGT and CVLT for all 3 configurations.
   • Set limits according to min and max 'radius of curvature' values noted in 'Comment' of surface No.4
8. Have optimization target defined by Optimization Wizard: E.g. RMS Spot Size
   
9. In the Multi-Configuration Editor: Set curvatures of Surface 4 variable.
   
10. Optimize the model. We suggest to first try a Local Optimization with a Damped Least Squares algorithm as shown in the screenshot below.
    
    

How to model typical optical aberrations into a liquid lens for a sensitivity check

The variable surface of liquid lenses typically shows some amounts of optical aberrations due to its flexible nature.

For liquid lenses mounted with a vertical optical axis, astigmatism (cylinder) is the main contributor to the overall wavefront error, which is typically specified in terms of “RMS wavefront error” in units of waves.

When using a standard liquid lens with a horizontal optical axis a gravity-induced coma term must be added resulting in a slightly increased total wavefront error. The gravity-induced coma term depends on the clear aperture diameter of the lens, the density of the liquid, the mechanical properties of the membrane and can be optimized upon request. Note that since 2023 Optotune provides gravity compensated (“GC”) lenses that make use of a secondary liquid to “fill up” the asymmetry caused by gravity, resulting in coma values well below 0.05 waves RMS.

The typical amounts of RMS wavefront error strongly depend on the liquid lens type, and Optotune continuously works on reducing them further. Please contact OPTOTUNE to request latest data.

In this section we provide an example of how to model typical optical aberrations into a liquid lens for a sensitivity check.

Let’s assume that Optotune specifies the maximum total wavefront error of a 12 mm clear aperture lens type with a vertical optical axis as 0.10 waves RMS. We can assume that the error fully consists of astigmatism (cylinder):

→ Wavefront_RMS_Astigmatism = 0.10 waves

To study the impact of such a wavefront aberration on the image quality of a system, we will physically model the corresponding surface aberration onto the variable surface of the liquid lens. For this we need to convert the RMS wavefront error to the corresponding RMS surface error (Zernike Standard Sag coefficient in units of mm) which has produced the wavefront:

→ Surface_RMS_Astigmatism (=Zernike Standard Coefficient 5 or 6) =

where

“WL” is the wavelength the wavefront was measured at => 0.53 um

“n” is the refractive index of the optical liquid e.g. of lens type EL-12-30 => 1.45

This value can now be used on a “Zernike Standard Sag” surface as the “Zernike 5” or “Zernike 6” coefficient. Let’s apply it to an EL-12-30 lens embedded in an imaging lens as shown below and check the impact on the imaging performance.

Apply the surface aberration to the variable surface:

1. Change the surface type of the variable surface from “Standard” to “Zernike Standard Sag”.
2. Set the “Maximum Term” of the Zernike expansion to 5
3. Set the “Norm Radius” of the Zernike expansion to 80% of the clear aperture semi-diameter of 6.0 mm => Norm Radius = 4.8 mm (this is because Optotune measures and specifies the Wavefront RMS Error over 80% of the lens’ clear aperture)
4. Set the “Zernike 5” coefficient (surface astigmatism) to the Surface_RMS_Astigmatism value calculated above.
   

5. Check the impact of the aberration on the imaging performance of the system:
   

The bottom row shows a slight decrease in the MTF performance compared to the nominal case.

Note: The Surface_RMS_Astigmatism  value calculated above can also be used with the TEZI operand in the Tolerance Data editor for a full sensitivity or Monte Carlo analysis. For this purpose, however, the surface type of the variable liquid lens surface must remain a “Standard” surface and not be changed to a “Zernike Standard Sag”.

For any questions about aberration modelling methods for these lenses, feel free to contact Optotune directly at sales@optotune.com.