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.