Internal Reflection of Magnifying Glass

A few months ago, I got myself a cheap 90 mm diameter magnifying glass in a local flea market and the idea was to make a simple refracting telescope out of it. Upon inspection, I realized there was an uneven distribution of its composition which causes anisotropy in its refractive index. This defect is probably due to improper cooling rate when the glass was poured into its mould.

Other than this, both surface of this lens have non-uniform radius of curvature which translates into distortion of image it creates. Spherical aberration is also expected and can be seen from the image it produce and lastly, due to its size, chromatic aberration is very much visible and therefore this lens is of poor quality which renders (pun intended) the initial high-resolution task impossible – something I should have expected for  material that cost only RM5.

The distortion and artefacts coming out from physical defects inside the magnifying class can be seen in the picture on top; as I prepared an example how a simple magnifying glass can be used as a crude form of telescope. By putting it in front of a camera, the magnifying glass becomes the objective lens of the telescope and the lens of my camera becomes the “eyepiece”. When you have a two lens system, a telescopic image can be produced on the detector of my camera in a fashion similar to the human retina. 

We note that the tree-like object is actually a telco transmission tower disguised as a tree. The poor image quality shown here is due to the physical defects mentioned earlier and chromatic aberrations are obvious as colours are dispersed around the “branches” of the “tree” especially near to the edge of the glass.

So the cheap lens was kept in the dry box for a while until I figured out the other day of using it as a demonstration for geometrical optics: with a layer of glass and a bright beam of light made possible by commercially available DPSS green laser, it is easy to show the effect of internal reflection, where the light beam is allowed through the magnifying glass and at the interface between glass and air, the beam touches the boundary at an angle exceeding the so-called critical angle which allows it to bounce back onto another interface and so on. 

The picture above clearly demonstrated internal reflection where a beam enters the magnifying glass from the right side as evident from the relatively large green spot (saturated by scattered light upon entrance into the glass) compared to other bright spots on the glass. This ray of light is then reflected inside the glass upon the glass-air boundary until the 6th order where the thickness of the magnifying glass is thinning upon the edge (think of the shape of a magnifying glass) and the internal reflection finally lost its critical angle upon the interface.

Although we can increase the number of times the beam gets reflected inside the lens, it should be noted that this kind of reflection has its loss mechanisms. That is the intensity of light after each reflection suffers reduction in its “brightness” due to transmission towards outside of the glass-air interface. The next picture illustrates how:

Six green streaks of light on the wall came from the bright points on the magnifying glass shows some part of the light gets out of the magnifying glass (instead of being internally reflected) and hit the wall at the back. The central hyperbolic-shaped shadow corresponds to the black opaque plastic which holds the magnifying glass. Notice the loss of light intensity is obvious as the brightness of the spots decreases when we count from left-bottom spot to right then to left again all the way up. In some ways, we can see each reflection point inside the magnifying glass is similar to the interface of a beam splitter where some parts of the light is being reflected into the interface and some being allowed to let through. 

Nonetheless, it should be noted that internal reflection is only applicable when the wave (in our case: in a beam of light) travel from a medium with a higher refractive index (in our case: the magnifying glass) to another medium (in our case: air) with a lower refractive index, and precisely because of this, internal reflection of waves found many applications in our life. 

One good example is the use of fibre optics in telecommunications, where digital signals are compressed in a form of light pulses and then sent into a sort of "pipe" that is made of glass or other transparent materials having a higher refractive index compared to its surrounding. This enable the light signals to internally reflect along the fiber which essentially behaves like a "light-conducting" cable, transporting important scientific signals or streaming YouTube videos across the continent. 

In the regards of use in architecture, the brilliant application of internal reflection manifested in light tubes like this. It is useful to reduce energy requirements for lighting (because natural lighting is dirt cheap if you don't consider the cost of light tube), thus making homes, offices or public buildings more environment friendly.

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Back to the demonstration: So previously, I placed the lens vertically on a wooden board allowing a beam of light coming from an adjustable plane. Now, assuming the optical axis of the magnifying glass were placed vertically parallel to the z-axis on the y-plane, introducing a beam of light from negative x, y, z axis towards the edge of the lens produced a fantastic internal reflection pattern where reflection order of 10 and above is observed and the path of reflection was bent according to the curvature of the lens. I reckon this reflecting configuration should have its practicality as a waveguide but I haven’t any idea where and how it should be used yet. 

Anyway, when the angle of incidence is adjusted carefully, the internal reflections take place in a ringed configuration suggesting the importance of curved surface geometry to the reflecting beam. Despite the appearance of high reflection order, we note the reflection spot in orders from 10 and above (at the top of the magnifying glass) starts to decay into a featureless smudge. 

Any good guess why?