APPENDIX
Calculation of the Power of an Intraocular Lens
This appendix calculates a mathematical expression that allows to know the paraxial power of an intraocular lens (IOL) from two measures in the experimental set-up: one of the system without IOL, and another one with it. For simplicity we assume that both the artificial cornea and IOL are thin lenses.
In the first step (without IOL), the cornea and the retina are present into the artificial eye:
Figure A1.
In Figure A1, a represents the distance from the entrance pupil plane (EP) to point O, whose image is located in the artificial retina, O'. O corresponds to the far point, and 1/a will be the refraction measured from the entrance pupil, whose value is provided by the aberrometer. The power of the cornea, Pc, can be calculated taking into account that:
𝑃𝑐 = 𝐴𝐿𝑛′ −𝑎1 Eq. A1
where n' is the index of the saline solution for the wavelength of the measuring light (780 nm), and AL is the axial length of the artificial eye. In Eq. A1, and the rest of calculations it is used a sign convention where a and AL are both positive. From Eq. A1 we can obtain the AL value knowing that Pwo=1/a:
𝐴𝐿 = 𝑃 𝑛′
𝐶+ 𝑃𝑤𝑜 Eq. A2
Figure A2 shows the artificial eye with the IOL embedded.
Figure A2.
Where s1(<0) is the distance from the object to the artificial corneal. Considering the thin lens approximation and taking into account that from Figure A2:
𝑠2 = 𝑠1′ − 𝑒 Eq. A3
𝑃𝑤 = 𝑠𝑛′
1′− 𝑃𝐶 Eq. A4
𝑠2′ = 𝐴𝐿– 𝑒 Eq. A5
where e(>0) is a known value that represents the distance between the artificial cornea and the IOL, and Pw is the total power of the artificial eye with IOL, and whose value is provided by the measurement of the aberrometer. From Eq. A3 and A4, we can obtain:
𝑠2 = 𝑃 𝑛′
𝐶+ 𝑃𝑤− 𝑒 Eq. A6
where e is a known value that represents the distance between the artificial cornea and the IOL, and Pw is the total power of the artificial eye with IOL, and whose value is provided by the measurement of the aberrometer by taking into account the value of the defocus Zernike coefficient, C20.
On the other hand, we know that paraxial IOL power for 780 nm, is given by the following expression:
𝑃780𝐿𝐼𝑂 = 𝑠𝑛′
2′ −𝑠𝑛′
2 Eq. A7
and, using equations A1, A2, A5, A6 and A7 and after some calculations, we can obtain the power of the IOL from the measured parameters:
𝑃780𝐿𝐼𝑂 = 𝑛′2[[𝑛′−𝑒(𝑃 𝑃𝑤𝑜−𝑃𝑤
𝐶+ 𝑃𝑤𝑜)][𝑛′−𝑒(𝑃𝐶+ 𝑃𝑤)]] Eq. A8
Finally, to calculate the IOL power for a wavelength centered in the visible spectrum, such as 550 nm that is normally used in the standards, it should be noted that:
𝑃𝜆 = (𝑛𝐿𝜆− 𝑛′𝜆)𝑘 Eq. A9
where n’λ and nLλ are the refractive indices of the saline solution and the IOL for a particular wavelength (known values) respectively; and k is a constant that depends only on the geometry
of the IOL. Using Eq. A9 for two wavelengths, 780 and 550nm, we obtain the following expression:
𝑃550 = 𝑃780(𝑛𝑛𝐿550−𝑛′550
𝐿780−𝑛′780) Eq. A10
Thus, the final expression for the IOL power for a wavelength of 550 nm can be obtained from Eqs. A8 and A10:
𝑃550𝐿𝐼𝑂 = 𝑛′7802[[𝑛′ 𝑃𝑤𝑜−𝑃𝑤
780−𝑒(𝑃𝐶+ 𝑃𝑤𝑜)][𝑛′780−𝑒(𝑃𝐶+ 𝑃𝑤)]] (𝑛𝑛𝐿550−𝑛′550
𝐿780−𝑛′780) Eq. A11
