Section 27–6 Aberrations

 (Spherical aberration / Chromatic aberration / Comatic aberration)

 

In this section, Feynman discusses spherical aberration, chromatic aberration, and comatic aberration (or coma).

 

1. Spherical aberration:

“This effect is called spherical aberration, because it is a property of the spherical surfaces we use in place of the right shape (Feynman et al., 1963, section 27–6 Aberrations).”

 

According to Feynman, a real lens having a finite size will exhibit aberrations, for example, spherical aberration is a smear in an image. The word spherical is used because spherical aberration is related to the spherical surfaces of the lenses that enlarge an image imperfectly. Simply phrased, an aberration is an image error of an optical system that may manifest as an unclear or distorted image. Spherical aberration is classified as a type of monochromatic (or quasi-monochromatic) aberrations, however, other types of monochromatic aberrations include coma, astigmatism, field curvature, and distortion. More important, one should emphasize that spherical aberration is observable even for objects that are located on the optical axis (or principal axis).

 

Feynman elaborates that the spherical aberration could be remedied by re-forming the shape of the lens surface or using several lenses arranged so that the aberrations of the individual lenses tend to cancel each other. One may include more methods to resolve this aberration such as using an aperture stop or computerized lens design. In short, the aperture stop can affect the amount of light closer to the optical axis that passes through the lens. Perhaps Feynman should mention that computerized lens design is a useful tool that was used by manufacturers worldwide in the early 1960s (when this lecture was delivered). Currently, there are more sophisticated computer programs that help to design and analyze more complicated optical systems (Hecht, 2002).

 

2. Chromatic aberration:

“So if we image a white spot, the image will have colors, because when we focus for the red, the blue is out of focus, or vice versa. This property is called chromatic aberration (Feynman et al., 1963, section 27–6 Aberrations).”

 

According to Feynman, another fault of the lens is its refractive index which is color-dependent, and thus, light of different colors travels at different speeds in a glass. That is, a white spot has chromatic aberration in the sense that its image has different colors. Chromatic aberration is also known as “color fringing” or “splitting of light” because the lens is unable to let colored light rays meet at the same point in the focal plane. Specifically, lens dispersion is observed as a result of a higher refractive index for light rays that have shorter wavelengths. In other words, the image appears blurred with colored edges because light of different colors reaches different points along the optical axis.

 

Instead of providing a specific solution to compensate for chromatic aberration, Feynman asks how careful do we have to be to eliminate aberrations. Then, he says that the theory of geometrical optics does not work here and the principle of least time is only an approximation. Perhaps Feynman should discuss how achromatic lenses can resolve the chromatic aberration from the perspective of Fermat’s principle of least time. For example, one may explain to what extent the light path of each ray has the same length by having a good design of lens surface with the appropriate refractive index. In essence, Fermat’s principle of least time is a first-order approximation in the sense that the optimum light path could have the longest-time or shortest-time provided all nearby paths take approximately the same time (δT = 0).

 

3. Comatic aberration: “If the object is off the axis, then the focus really isn’t perfect anymore, when it gets far enough off the axis (Feynman et al., 1963, section 27–6 Aberrations).”

 

Feynman says that the focus isn’t perfect anymore if the object is located off the optical axis. (Strictly speaking, the focus isn’t perfect even if the object is located on the optical axis depending on its size and colors.) He elaborates that the image will usually be quite crude, and there may be no place where it focuses well. One may clarify that this image error is also known as coma or comatic aberration. In short, coma is a monochromatic aberration that occurs for an object located a distance from the optical axis; the light rays reach points on the focal plane that are farther from the optical axis. A good example is the appearance of comet-shaped stars when they are located at an angle to the optical axis of a telescope.

 

Feynman mentions that the optical designer tries to remedy aberrations by using many lenses to compensate for each other’s errors. Then, he explains that if we have arranged the time difference for different light rays is less than about a period, there is no use going any further. One may add that comatic aberration can be compensated by using an aperture stop at the proper location. However, if the time difference for different light rays is less than about a period, then there would be interference between the light rays such that it is more difficult to improve the resolution of the image. This is a limitation of geometrical optics because of the diffraction and interference of light waves.

 

Review Questions:

1. How would you define the concept of spherical aberration?

2. How would you resolve the problems of chromatic aberration?

3. Do you agree with Feynman when he says that the focus isn’t perfect anymore if the object is located off the optical axis?

 

The moral of the lesson: aberrations (image errors) are due to the faults of lenses (spherical surface and refractive index) that cause light rays not to meet at the focal point and at the same time (Fermat’s principle of least time).

 

References:

1. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

2. Hecht, E. (2002). Optics (4th ed.). San Francisco, Addison-Wesley.

Section 27–5 Compound lenses

 (First principal plane / Second principal plane / Focal length)

 

In this section, Feynman discusses the first principal plane, second principal plane, and the focal length of an optical system of lenses.

 

1. First principal plane:

“If parallel light comes in the other way, it comes to a focus at the same distance f from the first principal plane, again as if a thin lens where situated there (Feynman et al., 1963, section 27–5 Compound lenses).”

 

Feynman states the property of first principal plane as “if parallel light comes in the other way, it comes to a focus at the same distance f from the first principal plane, again as if a thin lens where situated there.” In other words, if parallel light rays move from the last lens to the first lens of an optical system, they will meet at the focal point that is at the focal distance f from the first principal plane. The principal planes are idealized planes of the optical system whereby the light rays are assumed to refract. In the real world, the light rays would refract two times (instead of once) when it passes through the first lens at the two spherical interfaces. Thus, we can define the first (front) principal plane as an idealized plane through the intersections of the incident light rays (from the last lens) that are parallel to the optical axis and the emerging light rays (from the first lens) that meet at the front focal point.

 

Feynman explains that there is a very interesting net result of the effects of any sequence of lenses on light that starts and ends up in the same medium, say air. That is, he relates the effects of the compound lenses or optical system to the property of principal planes. However, the property of principal planes is a result of idealizations and simplifications: the light rays do not really bend at the principal plane and do not (exactly) meet at the focal point because of aberrations (see section 27–6). Physicists have simplified the paths of light rays within the optical system to provide a fairly accurate effect of the compound lenses. In general, we may apply the concept of principal plane to a thin lens, thick lens, or a system of lenses.

2. Second principal plane:

“If light comes into the system parallel from the first side, it comes out at a certain focus, at a distance from the second principal plane equal to the focal length… (Feynman et al., 1963, section 27–5 Compound lenses).”

 

Feynman states the property of second principal plane as “if light comes into the system parallel from the first side, it comes out at a certain focus, at a distance from the second principal plane equal to the focal length.” In other words, if parallel light rays move from the first lens to the last lens of an optical system, they will meet at the focal point that is at the focal distance f from the second principal plane. In addition, he says that the principal planes are close to the first surface of the first lens and the last surface of the last lens. However, the principal planes may lie inside or outside of the lens because the locations are dependent on the curvatures of the lenses’ surfaces and their thicknesses. We can define the second (back) principal plane as an idealized plane through the intersections of the incident light rays (from the first lens) that are parallel to the optical axis and the emerging light rays (from the last lens) that meet at the back focal point.

 

Feynman explains that the principal planes are coincident for a thin lens as if we could slice down a thin lens and not notice that it was separated. Strictly speaking, the principal planes are slightly curved, but it appears flat provided they are near to the optical axis (paraxial approximation). Furthermore, every ray that emerges from the first principal plane is assumed to move to the point at the second principal plane that has the same height. Note that there is a point-for-point correspondence between the two principal planes such that the object is equivalent to the image (at the principal planes) and both are of the same size. The principal planes are also called unit planes because the lateral magnification of the object at the first (or second) principal plane is +1. In other words, the two principal planes are idealized planes that have unit lateral magnification (without inversion).

 

3. Focal length:

“… the formula (27.16) that we have written for the thin lens is absolutely general, provided that we measure the focal length from the principal planes and not from the center of the lens (Feynman et al., 1963, section 27–5 Compound lenses).”

 

Feynman says that the formula xx' = f² for the thin lens is absolutely general, provided that we measure the focal length from the principal planes. In a sense, this is incorrect because a more general formula is xx' = ff'. Feynman has assumed that the refractive indices of the media in front of and behind the lens are the same, but this is not generally true. For example, the effective focal length of a lens in air or water is the distance of the focal point from the corresponding principal plane. Interestingly, the principal planes are characterized by a lateral magnification of +1 and this property is applicable to thin lenses, thick lenses, and a system of lenses. By using a more general Newtonian lens formula (xx' = ff'), the principal planes are unit planes because the lateral magnification Y = f/x = x′/f′ = +1 (Katz, 2002).

 

Feynman elaborates that the principal planes and the focal length may be found either by experiment or calculation, and we can use these two concepts to describe the properties of the optical system. Specifically, the relationship between the principal planes and the focal length can be derived by using the ray-transfer matrix that holds for the propagation of light rays between the two principal planes (Goodman, 2005). More important, the principal planes are conjugate planes that help to solve problems related to the focal lengths of a system. The two planes are conjugate to each other in the sense that if an object is in one of the two principal planes, its equivalent image is formed in the other principal plane. This can be shown by using the lens formula 1/x+1/x' = 1/f or xx' = f² where x and x' are clearly interchangeable within the formula.

 

Review Questions:

1. How would you define the concept of first principal plane?

2. How would you define the concept of second principal plane?

3. Do you agree with Feynman that the formula xx' = f² for the thin lens is absolutely general, provided that we measure the focal length from the principal planes and not from the center of the lens?

 

The moral of the lesson: the two principal planes (unit planes and conjugate planes) provide useful references from which we can measure the effective focal length of an optical system or optical instrument.

 

References:

1. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

2. Goodman, J. W. (2005). Introduction to Fourier optics, 3rd ed. Englewood, CO: Roberts and Company Publishers.

3. Katz, M. (2002). Introduction to geometrical optics. World Scientific: Singapore.