Curvilinear Distortions




   A major problem faced with zoom lenses is the curvilinear distortions or the barrel and pincushion distortions. This is a distortion where the lateral magnifications of the image produced is not constant. (Langford) This happens with point objects off the axis. (Monk p60) In other words a variation in magnification with axial distance. (Sears, p125) This does not affect the sharpness of the image but just the shape. What we want is the magnification

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so that we can eliminate this problem.


Lateral (transverse) magnification:

   This is the ratio y'/y where y' is the produced image height and y is the original image height. (fig2-3 monk p11)

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Barrel (negative distortion):

   If you have an image of a square the magnification at the center is greater then at its edges.(Langford)


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The magnification decreases with axial distance (Sears, p119).



Pincushion (positive distortion):

   The image of the square is less magnified at the center then the edges


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The magnification increases with axial distance (Sears, p120).


Calculations of distortion and where it comes from.
Relative distortion D is given by. This can be found by the definition of derivative

         

we take h as f(x) we get

      

or if y' = f(x + f(x)) and y = f(x) we get

    


Now we know by trigonometry that y = tan u if u is the angle of the ray that coming in and y' = tan u' where u' is the angle of the ray coming out.

By Taylor series expansion of tan we have


D varies with the distance away from the axis or the off-axis distance therefore D is caused by the higher order terms of this expansion, namely the third and fifth order.



since we only care about the third and fifth order terms of the derivative. The coefficient "a" gives the distortion. If a > 0 then we have pincushion distortion and if a < 0 then we have barrel distortion



The Cause of Distortions and the Solution


Barrel and pincushion distortions can be caused by a stop or aperture. An aperture is a hole or opening, this could be the boundary of the lens. A stop restricts the amount of rays coming into the lens.


Barrel distortion is caused when the stop appears before the lens.

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Pincushion is when the stop appears after the lens.

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A method used to fix distortion is using both the lenses which have pincushion and barrel distortion together. A stop is usually put in between the lenses.


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Reference: http://www.vanwalree.com/optics/distortion.html
Sears, ch5