Correction for Astigmatism

Eye ball

To understand how to correct astigmatism, we have to know more about the structure of eyes. In an eye ball, there's a cornea (C), a lens (L) with adjustable width controlled by a group of muscles atound it, and finally a place at the back called retina (R). It is a location where all light ray should reach. If some of the rays by some reasons couldn't reach the retina, then image we see will become fuzzy. Normal Eye Astigmatism Correction Lens Astigmatism where the cornea is in irregular shape which makes light couldn't fully fall onto the retina. Therefore, optometrist either fix the cornea to its regular shape by refractive surgery or make a correction lens altering the path of light in order to let ray fall onto the retina again. Astignmatism analysis by Transform Matrix Before we go straight to learn how to cure astigmatism by correction lens, we better know how to trace the rays by transform matrix. Since we learned that there are three types of matrix, they would then be our components or tools to build up the ray trace. For example, we would represent the rays in normal eye and astigmatism as following: = Here, we found two new symbols from the equation, they are 'y' and 'theata'. 'y' indicates the height of the ray from the horizontal axis, while theta the angle from the horizontal axis. The y and theta with an asterisk at the very beginning of the equation represent the new height and new angle of the ray. On the other hand, let M1 be the first matrix after the equal sign, M2 be the second and vice versa. So, M1 says the distance between the lens and the retina is 's' with index of refraction of the eye-ball fluid 'n'. M2, M3 and M4 represents the lens. M5 represents the distance between cornea and the lens. Finally, M6, M7 and M8 represent the cornea. The last matrix indicates the ray outside the eye. By the multiplication of these matrix, we can obtian where at the back of the eye-ball the ray would fall onto. This is a very useful data in determining which lens and how refraction surgery we should apply on the astigmatism patient. However, you may ask a question of whether we can simplify the calculation? This is a question a mathematician always ask for and the answer is "YES" in this case. Before we go to simplification, we have to make an assumption. It is the distance within the cornea and lens is very short. In other words, the cornea and lens are so thin that we can ignore the thickness. Thus, 's' is zero. This makes the translation matrix become an identity matrix. (i.e. A=D=1, B=C=0 ) Therefore, our new equation would be: = C = -1/f is the multiplication of two refraction matrixes, where f = (n2-n1)(1/R1-1/R2). R1 and R2 are the left and right refracting surface respectively. That means the R in M2 and M6 is R2, while the R in M4 and M8 is R1. Keep in mind that the above diagrams are only a bisection of an eye. But, at any angle we can view an eye from these bisection diagrams. With this key concept, we can further analyse astigmatism. Since the causes of astigmatism is due to the irregular shape of the cornea. This means there's a wrong value inside either M6, M7, M8 or all of them. This cue bring us a step further on how to tackle astignmatism and even some other eye-illnesses involved lens. Refraction surgery To allign with our assumption that cornea is a very thin substance of neglectable thickness, we may ignore M7 which is an identity matrix. This assumption leaves us two refraction matrixes, M6 and M8. We should first look at what variables involved in the matrixes and what they mean in our treatment. They are n1, n2 and R. 'n1' is the index of refraction of the fluid inside our eyes. 'n2' is the index of refraction of the media outside our eyes, which is air of index value 1. And the last one is R which is the curvature of our cornea. Obviously, we would not extract all the fuild out from our eyes and fill in another substance with an appropriate refraction index. And we cannot replace air with other things too. Thus, we may also neglect n1 and n2 which are the variables we cannot change. There are two R's left, one is the inside curvature of a cornea, and another one is the outside curvature. They are the only variables we may alter by refraction surgery. It alters the curvature of the cornea by using laser, which can only remove tissues from cornea. The extent of the tisses removed is determined by emmitting rays on the irregular part of cornea in order to see how far the rays apart from the retina. This is 'y', and surgeon would try to remove tissues to make it zero so that all rays could fall onto the retina again. (This is true, if we assume retina is horizontally parallel to the pupil, otherwise, we have to measure how far it's away from the pupil and set this value as the desired 'y'.) = This matrix is the multiplication result of the above matrix. The asterisks in the matrix are the values we may ignore in this case, since they do not affect the resulting 'y*'. To obtain 'y*' = 0, we have to make 's'='f'. Recall that f = (n2-n1)/(1/R1 - 1/R2). Thus, we may change the curvature that is R to alter 'f' in order to make it equals to 's'. Ha! This is our treatment to astignmatism by refraction surgery. Correction lens If you understand the idea behind refraction surgery, it's not hard to understand how correction lens resolve astigmatism. It is just place an appropriate lens in front of the eye-ball, like the following matrixes do. = = The last matrix is obtained by retrieving the * in the B's position of the pervious line matrix times the refraction matrix. It seems quite complicated, but the idea is simple and easy to solve. We always wants A=0, thus we have to solve nf^3 - f^3 - 2sf + s^2 = 0. This can be solved with only unknown f. Once we obtain f, we would then know the suitable curvature of the lens. This gives us the data to make an appropriate correction lens.

Barrel and pinchusion distortion

Chromatic Aberration

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