Aberrations

Natalie Gakopoulos

University of Arizona

Optical Science Center

nsophia@berkeley.edu

Abstract

Lens systems may produce distortions in the image called aberrations. There are many types of aberrations, categorized into 1^st order, 3^rd order, 5^th order, etc. The most common types of aberrations are the 3^rd order aberrations: chromatic, spherical, coma, field curvature, distortion, and astigmatism. These aberrations are discussed, and numerically generated examples are given.

Introduction

Aberrations are distortions of the image produced by the lens system. The wavefront aberration may be measured with the aid of a point-like source. The error is the optical path difference between the actual, aberrated wavefront and the ideal, reference wave front [4].

Theory

The function for a circular image is

circ(r/R)exp(i*Phi) (1)

where exp(i*Phi) is the phase of light transmitted through the circular aperture, Phi represents the aberration of the image, R is the radius of the circular aperture, and r is the radial coordinate in the initial field, equal to (x²+y²)^1/2. The x and y terms are the pupil coordinates. In an ideal situation in which there is no aberration, the exponential term goes to one, and the resulting image is simply a point. Figure 1 represents an ideal point with no aberration, generated from the computer program Matlab. However, most optical systems will face the issue of aberrations produced in the image. The general equation for an aberration is [2]:

Phi=W_IJKH^Ir^Jcos^K(theta) (2)

The coefficient W_IJK represents the magnitude of the aberration. The subscripts of W encode the exponents for the terms in the aberration. H is the height of the image, and &theta is the azimuth angle.

Fig. 1: An ideal image from a point source with no aberration present. The Matlab code used for this and all subsequent figures include constant values H=1 and W_IJK=10.2, and r is a function of x and y. The code used also sets rcos(theta) equal to x.

There are many types of aberrations. Common first order aberrations are wavefront tilt and defocus. Wavefront tilt means that the paraxial magnification is different from the actual system magnification. The corresponding equation for this is Phi=W₁₁₁Hrcos(theta). Defocus causes the actual image plane to be shifted from the paraxial image plane. This equation is Phi=W₀₂₀r².

The most common third order aberrations are chromatic, spherical, field curvature, distortion, coma, and astigmatism. Chromatic aberration includes longitudinal and lateral aberration along the axes. The velocity of light changes when it passes between different mediums. Short wavelengths travel slower in glass than long wavelengths. This causes the colors to disperse. A series of different colored focal points become arranged behind each other on the optical axis [1]. Chromatic aberration can be corrected by combining a positive (concave) low-dispersion lens with a negative (convex) high dispersion lens.

Spherical aberration is caused by the spherical shape of lenses. Blurring of the image occurs when the light rays passing through the lens are focused in slightly different places. It also causes the focal length of the system to vary with the radius of the pupil. Shifting the image plane from paraxial focus can help correct this. The equation for spherical aberration is Phi=W₀₄₀r⁴. Figure 2 is a Matlab representation of spherical aberration.

Fig. 2: An image of spherical aberration. The spherical shape of a lens causes the light rays to focus in slightly different places, producing a blurred image.

Curvature of field occurs when the image produced by a lens is focused on a curved plane, but the film plane is flat. The equation for field curvature is Phi=W₂₂₀H²r². Figure 3 is a Matlab representation of field curvature.

Fig 3: An image of curvature of field. Curvature of field occurs when the image is focused on a curved plane but the film plane is flat.

Distortion, yet another type of aberration, shifts the image position. The images of lines that meet directly in the origin appear straight, but the images of any surrounding straight lines appear curved [7]. There are two types of distortion: pincushion and barrel. In pincushion distortion, the lines at the edge of the image bow inward. In barrel distortion, the edges bow out. The equation for distortion is Phi=W₃₁₁H³rcos(theta). Figure 4 is a Matlab representation of distortion.

Fig. 4: An image of distortion. Distortion shifts the image position. The images of lines that meet directly in the origin appear straight, but the surrounding lines are distorted, either bowed inward or outward.

Coma is an aberration that occurs when the object is not on the optical axis. The light rays enter the lens at an angle theta relative to the axis. This causes the system magnification to vary with the pupil position. The image is distorted to resemble the shape of a comet [3]. The equation for coma is Phi=W₁₃₁Hr³cos(theta). Figure 5 is a Matlab representation of coma.

Fig 5: An image of coma. Coma causes the image to represent a comet shape because the object or light source is not on the optical axis. The coefficient W₁₃₁ was set to 20.2 in this calculation.

Astigmatism also occurs off-axis. The farther off axis the object is, the greater the astigmatism. The lens is unable to focus horizontal (tangential) and vertical (sagittal) lines in the same plane [5]. Instead of focusing rays to a point, they meet in two line segments perpendicular to each other. These are the sagittal and tangential focal lines. The light rays in these two planes are imaged at different focal distances. The resulting image includes a horizontal or vertical line of light [1]. The equation describing this aberration is Phi=W₂₂₂H²r²cos²(theta). Figure 6 is Matlab representation of astigmatism.

Fig. 6: This image represents astigmatism. Astigmatism occurs when the lens is unable to focus horizontal and vertical lines in the same place. This results in an image with either a sharp horizontal or a sharp vertical line [6]. The Matlab code used to produce this image uses H=1, W₂₂₂=10.2, and r is a function of x and y.

The equation for the combination of all possible aberrations in a system is:

W= W₀₂₀r²+ W₁₁₁Hrcos(theta) + W₀₄₀r⁴+ W₁₃₁Hr³cos(theta)+W₂₂₀H²r²+ W₃₁₁H³rcos(theta)+ W₂₂₂H²r²cos²(theta) + 5^th and higher order terms (3)

Correcting aberrations in optical systems

Many different types of photographic lens systems have been created to eliminate or reduce aberration. The first mathematically calculated lens design was the Petzval portrait lens, still in use today. [1]. It consists of two groups of lenses, achromatic lens pairs, separated by airspace. Achromatic lenses are two combined lenses, one positive and one negative and each with a different dispersion (different index). Petzvel lenses are useful in correcting spherical aberration and coma.

Another notable type of lens system is the Cooke triplet. This system consists of three lenses separated by air. The outer lenses are positive lenses made of flint glass, and the middle lens is a negative lens made of crown glass [1]. The importance of the Cooke triplet is that, according to Price, “it contains the smallest number of elements by means of which all seven of the third-order aberrations can be eliminated.”

Zoom lenses are systems of lenses that can vary focal length, by varying the spacing between the lenses. Because the system uses multiple, separated lenses, it is susceptible to flare. Flaring arises from highlights within the system of lenses. These highlights interact with the optical surfaces to produce sharp amounts of light that spread around the system. This problem can be corrected by adding a coating to the lenses, reducing reflection and increasing transmittance. Zoom lenses make use of lens coatings to diminish the flare effect.

Aberrations are also present in the human eye. Myopia and hyperopia are two of the more significant and common defects. They are caused by defocus. In myopia, the power of the eye is too large for its axial length. Images of objects in the distance form in front of the retina. This aberration can be corrected with a concave lens in front of the eye. Hyperopia is the condition in which the power of the eye is too weak for its axial length, the images form behind the retina. This can be corrected with a convex lens. Visual astigmatism is another problem that can be present in the eye. Astigmatism occurs when the cornea or lens is not symmetric, causing variations in the power of the eye [2]. This forms linearly blurred images (see Figure 6). However, visual astigmatism can be corrected with the use of a cylindrical lens. The cylindrical lens introduces conventional astigmatism most strongly in the direction of the maximum error of the eye to correct it.

Summary

Aberrations are inherent in lens systems, whether in photographic lenses or in the human eye. Understanding and analyzing the theory and calculations behind aberrations allows for the design of the most practical lens systems with the most favorable images. The most common aberrations are chromatic, spherical, coma, field curvature, distortion, and astigmatism. These aberrations can be calculated and represented by images from Matlab.

Sample Matlab Code for coma

clc

clear all

N=200;

R=4*sqrt(N/pi);

for x=1:N;

for y=1:N;

xnew=x-N/2-1;

ynew=y-N/2-1;

r=(xnew^2+ynew^2)^(1/2);

xgraph(x)=xnew;

ygraph(y)=ynew;

if r < R

matrix(x,y)=1;

else if r == R

matrix(x,y)=.5;

else matrix(x,y)=0;

end

end

%matrix(x,y)=matrix(x,y)*(1+i*10.2*1*(r/R)^2*xnew/R);

matrix(x,y)=matrix(x,y)*exp(i*20.2*1*(r/R)^2*xnew/R);

end

end

fftshift(matrix);

fftmatrix=fftn(matrix);

newmatrix=abs((fftshift(fftmatrix)));

maxim=max(max(abs((newmatrix))));

newmatrix=abs(255*(newmatrix)/maxim);

maxmin=max(max(newmatrix));

image(xgraph,ygraph,newmatrix)

colormap('gray')

References:

[1] Brandt, H.M. The Photographic Lens. London: The Focal Press, 1968.

[2] Greivenkamp, John E. Field Guide to Geometrical Optics. Bellingham: SPIE, 2004.

[3] “Lens Optics.” Wikipedia. November 2005 <http://en.wikipedia.org/wiki/Lens_%28optics%29>.

[4] Longhurst, R.S. Geometrical and Physical Optics 3^rd Ed. Hong Kong: Continental

Printing Co. Ltd, 1973.

[5] “Photographic’s Super Course of Photography: Photographic Lenses.” Petersen’s Photographic. 24 (1995): 63-79.

[6] Price, William H. “The Photographic Lens.” Scientific American. 72 (1976): 72-83.

[7] Wyant, J.C. “Basic Wavefront Aberration Theory for Optical Metrology.” Applied Optics and Optical Engineering. 6 (1992): 1-53.