Conditions under which two-element variable power lenses can be created. Part 1. Theoretical analysis.

Area of Science:

• Optical Engineering
• Lens Design
• Variable Power Optics

Background:

• Variable power lenses are crucial for applications requiring adjustable optical effects.
• Understanding the design constraints of multi-element variable power lenses is essential.
• Previous research has not fully defined the conditions for creating two-element variable power lenses with invariant functional forms.

Purpose of the Study:

• To investigate the conditions necessary for creating a two-element variable power lens.
• To define a variable power lens where the optical effect's functional form remains constant during element translation, only its magnitude changes.
• To explore both lateral and rotational translation methods for achieving variable optical power.

Main Methods:

• Analysis of optical effects generated by the translation of two lens elements.
• Mathematical modeling to determine the functional forms of optical effects achievable.
• Mapping of designs from Cartesian space to polar coordinates for rotationally translating lenses.

Main Results:

• Identified that only variable power optical effects describable by quadratic functions can be formed using laterally translating two-element lenses.
• Demonstrated that designs for rotationally translating lenses can be derived by transforming laterally translating designs into polar coordinates.
• Established the specific conditions and mathematical framework for creating such variable power lenses.

Conclusions:

• The functional form of optical effects in two-element variable power lenses is constrained by the translation method (lateral vs. rotational).
• Quadratic functions are the sole form for optical effects in laterally translating two-element variable power lenses.
• A clear method exists for designing rotationally translating variable power lenses by adapting lateral translation principles.