Stokes Vector Method for Combining Cylinders
When two thin cylindrical lenses are placed in contact with their axes at oblique angles, the resulting combination cannot be described as a simple cylinder. Instead, it produces a sphero-cylindrical system. The Stokes vector decomposition method converts each cylinder into three components for vector addition.
Each cylinder C at axis A is decomposed as: S0 = C (total power), S1 = C·cos(2A), S2 = C·sin(2A). The resultant components are summed: S0r = S01+S02, S1r = S11+S12, S2r = S21+S22. The resultant cylinder magnitude is Cr = √(S1r² + S2r²), and the axis is Ar = ½·arctan(S2r/S1r). The induced sphere is Sr = (S0r - Cr)/2.
Special Cases
When two equal cylinders are placed at axes 90° apart, they produce a spherical equivalent with zero residual cylinder. When axes are identical, the cylinders simply add algebraically. At oblique angles between 0° and 90°, the full Stokes computation is required.
Frequently Asked Questions
What is a Jackson cross cylinder?
A Jackson cross cylinder (JCC) is a special lens with equal and opposite cylinder powers on perpendicular axes (+0.25/-0.25). It is used during subjective refraction to refine the axis and power of astigmatic correction.
Can I use this for over-refraction calculations?
Yes. When performing an over-refraction over a toric contact lens, the trial lens cylinder and the contact lens cylinder are at different axes and must be combined using this crossed cylinder method.
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