Is refraction intended to be a valid explanation of apparent magnification in the traditional and electron microscopes? A consideration of Snell’s Law and de Broglie’s equation.
1.In traditional optics, how would refraction, or the bending of a light ray as it enters a different medium, account for an enlarged image? What is apparently being stretched in some way is the object, in the sense that although it is called an image it is a real one because we are viewing it directly, not a virtual image projected onto a screen or, for instance, captured within the lens. But is this conceivable, or does it ‘stretch’ credulity?
2. Is the term 'normal' supposed to lead us to think about the fact that even if light does bend and this has an effect on the size of the object/image we view, it would also bend back to the angle at which it entered the lens, presumably, as it left it? (And the use of the term ‘ordinary glass’ sometimes used with the unit of refractive index might be to imply that at least one of the lenses in the cylinder, the objective lens, is in fact just that, plane glass rather than a convex lens.)
3. The calculation used to find out the two different angles, before and after it changes medium, is Snell’s Law: sin(e) of the incident angle x refractive index = sin(e) of the refracted angle x refractive index. Refractive index is related to the speed at which light travels in a particular medium but inversely, so that it is higher for a lens than in air. If the angle of transmission, or refraction, bends towards the 'normal' - ie, towards the perpendicular - in the denser glass of the lens with traditional optics, how does this result in an enlarged image? By analogy, would a light projector that shone inwards rather than outwards as it left the projector result in a larger or smaller image on the screen ahead of it? If the light source is underneath the lens, as it is, for instance with the Apex Learner, then in what sense is the object or its image enlarged beyond the microscope? Would we not be seeing exactly the same image without the aid of the microscope?
4. However, with electron optics - which replaces light with electron beams and ordinary lenses with magnetic lenses - the variable in Snell’s Law is not refractive index but velocity itself. If the direction of the change of angle is the opposite that we would expect with traditional light refraction, should we really still consider what is taking place to be refraction? If we replaced a red light with a green one we would not expect the angle to increase rather than decrease. If the property of electron beams is such that a change of medium would produce the opposite effect to that of light this is not, as far as I have read so far, asserted or explained.
5. Would a difference in refractive index of only about 1.5 in traditional optics make enough difference in the angle to account for significant magnification? In other words, assuming the difference in angle were that of 60° and 90°, which would be the case if light approached the lens at 90°, would we then much more in the way of enlargement than a ratio of 3:2, assuming an effect of enlargement rather than reduction. (See point .. below.) If, however, we imagine a larger angle as it enters the lens, which is said to be the case with electron optics, then were the beam to reach the glass at 60° and it refracted to 90° , then would the result be diffraction or an image infinitely large? And what if the difference in refractive index were not 1.5 times but the 1000 times said to be possible in electron optics?
6. I have so far set to one side the fact that the variable was sine of the angle rather than the angle itself. However, although a correct calculation leads to variations the numbers are not extremely significant. For example, an angle of 30 in air will result in an angle of approximately 20 in the lens, which is almost the same as we would expect if we were doing the calculation with angles rather than sine.
0.5 x 1 = 0.33 x 1.5.
If we replace the 30 beam of light with one approaching the lens at 90, however, the equation is the following:
1 x 1 = 0.67 x 1.5,
Whereby the refracted angle, however, is approximately [45], not 60. While the change in the angle is proportional to the increase in sine, the difference is greater than if we were using the angle alone. However, since the difference is greatest at 90 [check] and is of a ratio of only 2:1 rather than 3:2, the difference in terms of enlargement is not significant if one considers the stated maximum enlargement in traditional optics is 800x or more. [1 more]
7. Of greater importance, why would sine be introduced other than to put off the layperson or to demonstrate that the equation is nonsense? Trigonometry is apparently necessary to calculate sides of a triangle with respect to an angle but not the ratio of two angles where a non mathematical variable is introduced, refractive index. In most cases, also, neither will it be a case of contiguous [?] triangles since the light will rarely reach the lens at a point, but this is beside the point when the non-mathematical or applied-mathematical variable of refractive index is the second of the two variables in the equation.
8. While we might expect Snell’s Law to be the same in traditional and electron beams, would it, however, even be applicable in electron optics given that we are talking about electron beams rather than light. Refractive indices are derived from the variations in speed of light in different media. Could we necessarily assume the variations to be the same with electron beams? On the other hand, neither can the replacement of refractive index with velocity in the equation, an entirely different matter, be deduced from the replacement of light with electron beams. With electron optics there is an independent variable, acceleration, on the basis that electron beams can apparently be accelerated or decelerated, whereas light beams, whatever their source, remain at a constant speed in a given medium. However, variations in speed do not imply that the angle should decrease at a lower speed - because of a higher refractive index - rather than increase.
9. With electron optics there is an independent variable, acceleration, on the basis that electron beams can apparently be accelerated or decelerated, whereas light beams, whatever their source, remain at a constant speed in a given medium.. The electron beam is accelerated - although not beyond the speed of light - or decelerated, apparently, to produce variations in refractive indices of up to now 1,000 times. However, because the change in velocity is said to be 'continuous' rather than abrupt, again can this be considered refraction at all?
10. Again with electron optics, most of the light is described as ‘scattering incoherently’ rather than bending at a precise angle as it enters the lens. Again, is this what refraction predicts?
11. Sin(e), used to calculate the incident and refracted angle in Snell’s Law, is calculated by dividing the length of the opposite side by that of the length of the hypotenuse ('sense of humour'?). Sin(e) increases from 0 - 1 as the angle increases. Why sin(e) is included when we are not concerned with sides and there is no triangle, I am not sure, although it will not affect the difference in the angles one way or another, since the numbers are still the derivative of angles.
12. If refraction is said to be the mechanism of enlargement - rather than convexity and concavity, which however feature in diagrams of what happens in a lens - then why is there not magnification when we look through any plane glass, such as a window?
13. Why is refraction considered the mechanism of magnification, in any case, when with a magnifying glass convexity and concavity might plausibly account for magnification and this is suggested by diagrams explaining magnification? However, even if refraction was not thought to be a necessary factor, you would still need to take it into account if refraction is something that happens to light anyway; and yet in diagrams of what happens to light as it enters lenses, a light ray travels through the centre point of the lens without bending at all. However, very large differences in refractive indices are said to be the principal mechanism of magnification with electron optics and the fact remains that it is said to be the mechanism in traditional optics, so that invalidating refraction will also invalidate magnification.
14. Waves are important in optics because they reduce clarity (the ability to distinguish an object) and resolution (the ability to distinguish between objects): light travels in waves and these produce diffraction, or blurring, rather than the precise directions we are said to need when refraction produces magnification. I have so far got stuck on the equation for wavelength, not yet being able to understand the difference between the variables, h and p. (Texts I have consulted can have a habit of doing that, explaining - I assume deliberately and obviously - what one of the variables is but not the other, which in the case of this journal article you go back a few pages for and then it seems to be the same thing.) According to an entry in the 1945 Journal of Physics, wavelength is equal to h over p, momentum, where both h and p appear to be momentum, or velocity x mass).
“It is well known that a theoretical limit exists for the resolving power of an optical instrument. The smallest object that may be perceived through a microscope is limited, owing to diffraction effects, by the wavelength of the light employed. Thus while light may be considered as traveling in straight lines in many cases, the wavelike properties of light must be considered in attempting to detect very small objects. The possibility that particles should also exhibit wavelike properties was first suggested by de Broglie (see paper I). He proposed to associate with a particle of momentum p a wavelength
ƛ = h/p.”
This might explain the use of the term ‘standard wavelength’ elsewhere: ie, that according to this calculation, wavelength would always be 1. The equation was apparently proposed by de Broglie. Although the journal author appears to be deliberately introducing confusion between light and electron beam waves and that of particles that might instead be referring to the objects being viewed, he then tells us that the same equation was used by Davisson and Germer with respect to electron waves. The author also makes a link between de Broglie’s equation and Dirac’s,used to calculate the different numbers of electrons travelling at different speeds within an electron beam although the ‘thus here also’ where de Broglie’s follows Dirac’s might be taken to be ironic, at least in the ease with which it might be applied.
Waves are important in optics because they reduce clarity. However, in any case, why would the shorter wavelengths said to be associated with electron beams rather than light produce less, rather than more, diffraction? An analogy, although it might be mixing metaphors inappropriately, would be that it is easier to see one's reflection in water where there are fewer ripples than in one where there are more.
15. An entry in the American Journal of Physics mentioned 'intervening media', which - setting aside any other meaning of media intervention - led me initially to think, instead of the light or electron beam in fact traveling straight from the air to the glass of the lens or to the magnetic lens do the bits of metal and plane glass as well as air or vacuum in between the air and lens need to be taken into account when calculating refraction? However, ‘on reflection’, the reference to intervening media is more likely to be ironic in that if, for instance, one takes the eyepiece off the Apex Learner microscope one sees nothing until what looks like a bright light at the end of the objective lens tube. The 'condenser' referred to in textbooks is not apparent.
16. This quotation from a 1958 American Journal of Physics article is one of many hints that what is going on is reflection rather than refraction:
"On viewing a bright point of light through the double mirrors, multiple images of the source are seen by the eye. The angle between successive images is twice the angle between the mirrors, irrespective of the angle of incidence."
In fact, this might be suggesting that the larger image is the result only of a larger eyepiece lens compared to the size of both the objective lens tube and the circle that the object is placed on - with variation being accounted for by whether or not the screen is filled as well as by external interference and manipulation of the image.
17. Refractive index is derived from the velocity of light in the media, which is derived from what? It might be thought the other way round if one uses a protractor to measure the incident and refracted angles - an exciting new development in the measurement of Snell’s Law according to an entry in the 1958 edition of the American Journal of Physics.
Bibliography
Although it might be thought I have relied too heavily on out-of-date texts, they seem to make more obvious hints than later works, including A level textbooks, that I have consulted.
American Journal of Physics, 1945 and 1958 editions.
Hecht, Eugene. Outline College Physics, 2012.
el Kareh and el Kareh, Electron Optics, 1970.
Wikipedia entries on: refraction, the electron microscope, magnetic lenses, Dirac’s Law, momentum.
