Section 27–4 Magnification

 (Lateral magnification / Longitudinal magnification / Angular magnification)

 

In this section, we can find mathematical expressions that are related to lateral magnification, longitudinal magnification, and angular magnification.

 

1. Lateral magnification:

“Equation (27.15) is the famous lens formula; in it is everything we need to know about lenses: It tells us the magnification, y′/y, in terms of the distances and the focal lengths (Feynman et al., 1963, section 27–4 Magnification).”

 

Feynman defines the magnification of an optical system as y′/y that is in terms of an object height and image height as well as using xx′ = f² that is in terms of extrafocal distances (x and x′) and the focal length. However, the optical system is idealized to have only a focal length (instead of focal lengths: xx′ = ff′) and no aberration (see section 27.6). Feynman could have distinguished the concept of magnification using the terms lateral magnification and longitudinal magnification. Specifically, we can define lateral (or transverse) magnification as the ratio of the image height to the object height of the optical system. The lateral magnification would have a negative value if we consider the image is inverted with respect to the object.

 

In order to understand the formation and magnification of an image better, Feynman state two useful facts: 1. parallel rays from one side of the lens would proceed toward the focal point that is at a distance f from a lens. 2. any ray that passes the focal point on one side of a lens will emerge parallel to the optical axis on the other side. Importantly, there is one more useful fact: any ray that passes through the optical center of the lens will continue to move in a straight line. Furthermore, the first two conditions are applicable to a thick lens by including the concept of principal planes. This concept is also applicable to an optical system that has more than one lens (see section 27.5).

 

2. Longitudinal magnification:

“We leave it to the student to demonstrate that if we call s = x+f and s′ = x′+f, Eq. (27.12) is the same as Eq. (27.16) (Feynman et al., 1963, section 27–4 Magnification).”

 

Feynman mentions that y′/y = x′/f = f/x Eq. 27.15 is the famous lens formula and it has everything we need to know about lenses. Historically, the concept related to xx′ = f² appeared in Newton’s Opticks (Hecht, 2002) and thus, it is known as the Newtonian lens formula. Strictly speaking, the equation does not contain everything about lenses (e.g., aberration) and Feynman could have clarified how the lateral magnification of a lens is connected to the image distance s′ and object distance s (as well as x′ and x). Importantly, it is not trivial to explain that the extrafocal distances x and x′ are measured from the principal planes instead of the optical center. Furthermore, it is beneficial to explain that the longitudinal (or axial) magnification X is defined to be equal to x′/x (or X = Dx′/Dx = s′/s).

 

Feynman leaves it to his students to demonstrate that s = x+f and s′ = x′+f Eq. 27.12 are the same as xx′ = f² Eq. 27.16. However, it is worthwhile to let students demonstrate the concept of longitudinal magnification by differentiating the Newtonian lens formula. Firstly, we may express xx′ = f² as x′ = f²/x and differentiating x′ with respect to x, we have Dx′/Dx = f²(-x^-2) = -f²/x². Secondly, using xx′ = f² and y′/y = x′/f = f/x Eq. 27.15, we can get x′ = f(y′/y) and x = f/(y′/y). Thus, x′/x = f(y′/y) ¸ f/(y′/y) = (y′/y)² and it means that the longitudinal magnification X = x′/x is equal to the square of lateral magnification Y = y′/y. In a sense, the longitudinal magnification is about to what extent the image “becomes” larger because it appears farther.

 

3. Angular magnification:

“Now we may derive a lens formula. Using the similar triangles PVU and TXU, we find y′/f = y/x (Feynman et al., 1963, section 27–4 Magnification).”

 

To derive the Newtonian lens formula, Feynman uses the similar triangles PVU and TXU to get y′/f = y/x Eq. 27.13 and from similar triangles SWR and QXR to get y′/x′ = y/f Eq. 27.14. However, the angular magnification of a magnifying device is defined as M_a = q′/q or M_a = tan q′/tan q in which tan q = y/x (Note: y/x appears in Eq. 27.13). Simply phrased, angular magnification is the ratio of the angle subtended by an image (using a magnifying device) to the angle subtended by an object (naked eye). This is an important concept that is used to specify the magnification of telescopes and microscopes. (Feynman briefly mentions telescopes and microscopes in the next section without further elaboration.)

 

Some may understand the concept of magnification better by comparing lateral magnification, longitudinal magnification, and angular magnification. In short, we can define angular magnification, lateral magnification, longitudinal magnification as M_a = tan q′/tan q = (y′/x′)/(y/x), Y = y′/y, and X = x′/x respectively. The three magnifications can be related as M_a = (y′/x′)/(y/x) = (y′/y)/(x′/x) = Y/X. That is, we have Lateral magnification = Longitudinal magnification × Angular magnification (Y = X × M_a). On the other hand, it can be shown that the angular magnification of a lens is the reciprocal of its linear magnification (M_a = 1/Y). By assuming the object is small and it is placed on the optical axis, we have M_a = tan q′/tan q = (h/s′)/(h/s) = 1/(s′/s) that is valid for paraxial rays (Landsberg, 2000).

 

Review Questions:

1. How would you define the concept of lateral (or linear) magnification?

2. Does the famous lens formula contain everything we need to know about lenses, such as longitudinal magnification?

3. How would you explain the concept of angular magnification?

The moral of the lesson: the Newtonian lens formula xx′ = f² tells us the magnification, y′/y, in terms of the extrafocal distances measured from the focal points and the focal length of a magnifying device.

 

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.

3. Landsberg, G. S. (2000). Textbook of Elementary Physics, Volume 3 (A. Troitsky, Transl.). Honolulu, Hawaii: University Press of the Pacific. 

Section 27–3 The focal length of a lens

 (Spherical interface / Thick lens / Thin lens)

 

In this section, Feynman discusses the formulas of a spherical interface, thick lens and thin lens.

 

1. Spherical interface:

“Most of the lenses that we use have two surfaces, not just one. How does this affect matters? (Feynman et al., 1963, section 27–3 The focal length of a lens).”

 

Feynman says that we need a generalization of the formula (1/s) + (n/s′) = (n−1)/R (27.3) for a case where there are two different indices, n₁ and n₂, rather than only n. He adds that it is not difficult to prove that the general form of (27.3) is (n₁/s) + (n₂/s′) = (n₂−n₁)/R. In fact, the general formula can be proved using the formula (27.2) (h²/2s) + (nh²/2s′) = (n−1)h²/2R. One should realize that the formula (27.3) is simplified in the sense that the refractive index of the first term is 1 and the second term is n. By replacing 1 by n₁ and n by n₂, the formula (27.2) becomes (n₁h²/2s) + (n₂h²/2s′) = (n₂− n₁)h²/2R, and we have almost obtained the general formula.

Feynman elaborates that most of the lenses have two surfaces instead of only one. It is based on his broader definition of a lens and that is why the title of this section is “The focal length of a lens.” Specifically, the section is about the focal length of a spherical interface, thick lens, and thin lens. The title “the focal length of a lens” is potentially misleading because we tend to conceptualize a lens to have two surfaces. In general, a lens is defined as an optical device that comprises two refracting surfaces separated by a transparent medium (e.g., glass) and it can either converge or diverge light rays. Thus, the first and simplest situation could be more appropriately described as a spherical interface instead of a lens (that has one refracting surface).

 

2. Thick lens:

“…Therefore the thickness of the lens at the center must be given by the relationship (n₁h²/2s) + (n₁h²/2s′) = (n₂−n₁)T (27.8) (Feynman et al., 1963, section 27–3 The focal length of a lens).”

 

To derive the thickness of a lens, Feynman explains that the excess time in going from O to O′ is (n₁h²/2s) + (n₁h²/2s′) by considering the distances from O to P and P to O′ (diagonal distances) and initially ignoring the thickness T of a lens of index n₂. Then, he expresses the thickness of the lens T in terms of the radii R₁ and R₂ of the two surfaces as T = h²/2R₁ − h²/2R₂ (horizontal distances). The term h²/2R₁ is the horizontal distance from the vertex (or pole) of the first refracting surface to its center of curvature, whereas the term h²/2R₂ is the horizontal distance from the vertex of the second refracting surface to its center of curvature. However, one may define a thick lens as an optical device having two spherical refracting surfaces separated by a transparent medium that has a thickness T. One should realize that Feynman uses a special case (converging meniscus lens) to derive a general formula of a thick lens.

Feynman suggests that the thickness of the lens at the center must be given by the relationship (n₁h²/2s) + (n₁h²/2s′) = (n₂−n₁)T (27.8). Perhaps he could have clarified the formula n₁h²/2s) + (n₁h²/2s′) = (n₂−n₁)T is derived using the optical path, where the formula T = h²/2R₁ − h²/2R₂ is based on actual distance using only geometrical considerations. However, a better formula for thick lens can be expressed as n/f = n′/f₁′ + n′′/f₂′′ - dn′′/f₁′f₂′′ = n′′/f′′ (Gaussian formula) or P = P₁ + P₂ – dP₁P₂/n′ (Power formula) in which n, f, P, and d refer to refractive index, focal length, power, and the thickness of a thick lens respectively (Jenkins & White, 1981). These two formulas can be derived using similar triangle methods that are based on how light rays passes through the front focal point and rear focal point of the thick lens.

 

3. Thin lens:

“…if one of the points is at infinity, the other will be at a point which we will call the focal length f. The focal length f is given by 1/f = (n−1)(1/R₁−1/R₂)… (Feynman et al., 1963, section 27–3 The focal length of a lens).”

 

During Feynman’s derivation of the lensmaker’s equation, he mentions that if one of the points is at infinity, the other will be at a focal point given by the equation 1/f = (n−1)(1/R₁−1/R₂). Note that the equation relates the focal length of a thin lens to a ratio of refractive indices n = n₂/n₁ and the radii of curvature of the two lens surfaces. Perhaps Feynman could have emphasized the need of using the spherical refracting surface equation (n₁/s) + (n₂/s′) = (n₂−n₁)/R two times. Essentially, the image formed by the first refracting surface acts as an object (source of light) for the second refracting surface. Furthermore, the equation involves an idealization in which the lens is very thin in the sense that the separation between the two refracting surfaces is negligible compared with the object distance and image distance.

 

According to Feynman, it would be better to write the equation in terms of the focal length directly as (1/s) + (1/s′) = 1/f. In addition, the two focal lengths of an optical system are the same provided the initial and final refractive indices are the same. Curiously, one may observe that the lensmaker’s equation 1/f = (n−1)[(1/R₁−1/R₂)] does not include the size of the object. That is, one limitation of the equation is that the object size should be sufficiently small in comparison to R₁ and R₂. Furthermore, one may add that the lens is usually used where the refractive media on both sides of the lens are 1 (air), and thus, the symmetry of the foci F and F' would not be violated.

 

Review Questions:

1. Would you adopt Feynman’s definition of a lens that may have one or two surfaces?

2. Would you use Feynman’s method to derive the thickness of a thick lens?

3. How would you explain the idealization or limitations of the lensmaker’s equation?

 

The moral of the lesson: an important principle of a lens is to use the position of an image that is formed through the first refracting surface to find the new position of the image that is formed through the second refracting surface of the lens.

 

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. Jenkins, F. A., & White, H. E. (1981). Fundamentals of Optics (4th Ed.). Singapore: McGraw-Hill.

Section 27–2 The focal length of a spherical surface

 (Spherical surface / Plane surface / Spherical mirror)

 

In this section, Feynman discusses the formation of an image through a single spherical surface, plane surface, and spherical mirror.

 

1. Spherical surface:

“The farther ones may deviate if they want to, unfortunately, because the ideal surface is complicated, and we use instead a spherical surface with the right curvature at the axis. (Feynman et al., 1963, section 27–2 The focal length of a spherical surface).”

 

Feynman says that the simplest situation is a single refracting surface and we desire to arrange the curved surface such that every ray from O which hits the surface, at any point P, will be bent to reach the focal point O′. That is, the time it takes for the light rays to go from O to P plus n×O′P is equal to a constant independent of the point P. However, the curved surface is a fourth-degree curve sometimes known as the Cartesian ovals (or oval of Descartes) that may be used for lens design. We may derive the curved surface using simple analytic geometry based on the equation OP + n×O′P = Ö[(x – o)² + y²] + nÖ[(x – o′)² + y²] = c in which o and o′ are the x co-ordinate of O and O′ respectively. The fourth-degree curve can be obtained after expanding the equation Ö[(x – o)² + y²] + nÖ[(x – o′)² + y²] = c.  

 

According to Feynman, the law that connects the distances s and s′, and the radius of curvature R of a single spherical surface is (1/s) + (n/s′) = (n−1)/R. The main point of the equation is: the excess time (due to extra path) along route OP (h²/2s) and the other route (nh²/2s′) is balanced by the excess time (due to extra refractive index) needed for (n−1)VQ = (n−1)(h²/2s). The equation derived is known as the Gaussian formula for a single spherical surface. We can also derive the formula (27.3) by assuming the angles of incident light rays are so small such that the sines of angles can be replaced by the angles (in radians). In section 27-3, he adds that most of the lenses have two surfaces, but this is not a common definition of a lens.

 

2. Plane surface:

“… if we look into a plane surface at an object that is at a certain distance inside the dense medium, it will appear as though the light is coming from not as far back (Feynman et al., 1963, section 27–2 The focal length of a spherical surface).”

 

In the case of a plane surface, Feynman let R equal to infinity and obtain (1/s) + (n/s′) = 0. It can be simplified as s′ = −ns, which means that if we can look from a dense medium into a rare medium and see a point in the rare medium, the image appears to be deeper by a factor n. When we look at the bottom of a pool from above, it is not exactly correct to say that the apparent depth of the pool is ¾ of the real depth just because the reciprocal of the refraction index of water is ¾. Specifically, the apparent depth would vary with the angle of incidence or any point in the pool with respect to the location of an observer’s eye. The factor of ¾ is approximately correct for paraxial rays that are small enough if the cosine of the incident angle is almost equal to one or the sine of the incident angle is almost equal to the same incident angle (in radians).

 

To illustrate the meaning of negative s′ (or negative image), Feynman uses a ray diagram (Fig. 27–3) whereby the light rays which are diverging from O, will be bent at the refracting surface, and they will not come to a focus. That is, O is too close to the refracting surface such that the image is formed “beyond infinity.” Feynman explains that the image is a virtual image if the rays appear to come from a fictitious point that is different from the original location. Although the virtual image cannot be directly observed on a screen, it is still possible for the light rays to converge and form an observable image if a suitable lens is chosen. On the other hand, a real image is not only observable by using a screen, its aerial image through the lens axis (or optical axis) can be seen directly by naked eyes.

 

3. Spherical mirror:

“… we leave it to the student to work out the formula for the spherical mirror, but we mention that it is well to adopt certain conventions concerning the distances involved s (Feynman et al., 1963, section 27–2 The focal length of a spherical surface).”

 

According to Feynman, the object distance s is positive if the point O is to the left of the surface, whereas the image distance s′ is positive if the point O′ is to the right of the surface. In addition, the radius of curvature of the surface is considered positive if the center is to the right of the surface. However, there should be clarifications on the freedom of adopting certain conventions in optics. For example, we have the freedom to let the object distance to be positive or negative if the object O is to the left of the spherical mirror. On the other hand, we also have the freedom to let the image distance to be positive or negative if the image O′ is to the right of the spherical mirror. Thus, we have at least four conventions based on the object distance and image distance.

 

Feynman mentions that if we had used a concave surface, our formula (27.3) would still give the correct result if we merely make R a negative quantity. The convention is commonly known as “Real is Positive” convention, but some students may not be able to determine the location of an image whether it is at the left or right of the spherical mirror. To have a better understanding of the nature of the image, we can use ray diagrams to show whether the image is upright or inverted. One may explain that the concave mirror behaves like a convex lens in the sense that the light rays refracted to the right, are instead reflected to the left. This is possible provided the concave mirror is small compared with its radius of curvature such that parallel rays are focused to a common point (or focal point).

 

Review Questions:

1. How would you explain the derivation of the formula of a single spherical surface, (1/s) + (n/s′) = (n−1)/R?

2. How would you explain the apparent depth of a swimming pool does not look as deep as it really is?

3. What are the pros and cons of “real is positive” convention for a spherical mirror?

 

The moral of the lesson: the equation (1/s) + (n/s′) = (n−1)/R can be explained by how the excess time of two diagonal rays (due to extra path) is balanced by the excess time of a horizontal ray (due to extra refractive index).

 

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.