The critical angle lens reflector has commercial applications that supersede ordinary mirrored reflectors. The physics of the critical angle lens reflector are based on the optics principle of total internal reflection. Total internal reflection is characterized as follows: (1) as shown in Figure 2, the light rays are reflected off of the reflecting surface whereby the angle of reflection – with respect to the perpendicular to the reflecting surface – is equal to the angle of incidence of the ray, and (2) the reflection of the light ray takes place with no loss of energy upon reflection.

Total internal reflection is an advantageous condition for the reflection of light because it is reflection without energy or heat loss. Total internal reflection has practical applications in fluorescent microscopy, optical emission spectrometers, and ordinary prism reflectors that are used in binoculars and field glasses.

Ordinary silver-mirrored surface type reflectors – found in typical flashlights, automobile headlamps, residential and architectural lighting fixtures – similarly reflect light adhering to the same principle as above whereby the angle of reflection is equal to the angle of incidence. However, compared to total internal reflection, the distinct difference here is a loss of light energy during the reflection process. This light energy loss is manifest in the absorption of heat energy in the reflecting surface. These reflectors are energy absorbing devices that diminish reflecting power, generate unwanted residual heat loss, and put a drain on air conditioning resources in commercial lighting fixtures.

The critical angle lens reflector described here is a solid, lens-shaped object that is fabricated from a visibly transparent material such as glass, plastic, silicone or epoxy. The reflector would be typically used in a practical application in a surrounding transparent medium – such as in air, liquid, solid or gas. The lens shaped surface of the reflector has that particular geometrical profile such that light rays – emanating from an interior light source – strike the interior reflector surface with a preferred, fixed incident angle whose magnitude is larger than the optical critical angle. From the basic principles of geometrical optics in physics, the optical critical angle is that particular angle whereby incident light rays that strike the reflecting surface – with an incident angle that is larger than the critical angle – will undergo total internal reflection.

The optics-physics behind critical angle reflectivity is incorporated in Snell's law illustrated in Figure 1.

Figure 1 shows a light ray incident upon a water-air interface. The index of refraction of the water is n1 and the index of refraction of the air is n2 where, for total internal reflection to occur, n1 > n2. The incident ray makes an incident angle θ 1 with the vertical (normal line) and the refracted ray makes a refracted angle θ 2 with the vertical (normal line).

Snell's law shows the connection between the incident and refracted angles and the corresponding indices of refraction of the two media in the system – n1 for water and n2 for air where n1 > n2. As the incident angle is increased, then, for the case where n1 > n2, the refracted angle θ 2 approaches 90 degrees. At the point where angle θ 2 equals 90 degrees, there is no longer a refracted ray that crosses the boundary interface into the air medium and the corresponding incident angle in medium 1 is called the “critical angle” denoted as θ c. Mathematically, at θ 2 = 90 degrees, since sin (90) = 1, then from Snell's law, the magnitude of the critical angle θ c is given by

Some typical values of critical angles θ c for various material interfaces for n1 > n2 are:

For a water-air interface, n1 = 1.33, n2 = 1.00, θ c = arcsin (1/1.33) = 48.6°

For Lucite-water interface, n1 = 1.50, n2 = 1.33, θ c = arcsin (1.33/1.50) = 62.7°

For a Lucite-air interface, n1 =1.50, n2 = 1.00, θ c = arcsin (1/1.50) = 41.8°

The lens shape of the critical angle lens reflector has the geometrical profile that forces the condition that light rays – emanating from an interior light source – strike the interior surface interface with a fixed incident angle φ that is larger than the critical angle of the system as defined by the two component materials at the interface. The cross-section geometrical profile of the lens-shaped surface has the shape of a curve whose coordinate points (x, y) are determined by the mathematical differential equation shown below:

This equation is valid over the two-dimensional domain x ≥ 0, y > 0. In this equation, (x, y) are the coordinate points on the geometrical profile curve, dy/dx is the slope (derivative) of the curve and φ is the preferred, fixed angle of incidence of the light ray emanating from the light source. The three-dimensional profile of the preferred reflecting surface is a revolution of this two-dimensional curve about the symmetry y-axis.

In a practical implementation of this equation, the value of the preferred, fixed angle of incidence φ must first be declared in order to achieve total internal reflection of the light rays. This angle is determined by the material from which the lens is fabricated and the medium which surrounds the lens. As an example, we chose a practical system interface whereby the lens reflector material is clear Lucite plastic and the surrounding medium is air. From the above calculation for this particular system, the critical angle was evaluated to be 41.8 degrees.

Therefore, to get total internal reflection, the preferred, fixed incident angle must be larger than 41.8 degrees. We choose the preferred, fixed incident angle to have the magnitude φ = 45 degrees. For this example, the above equation for the two-dimensional geometrical profile configuration curve of the surface then reduces to the differential equation:

for x ≥ 0, y > 0. This differential equation has a unique solution for the given practical boundary condition y(0) = 1. Figure 2 shows the two-dimensional solution curve to this equation in the right-hand plane region x ≥ 0, y > 0.

Figure 2 shows a single ray coming from the coordinate origin point (0,0) and striking the reflecting surface at the particular coordinate point (x, y). The tangent line and the normal (perpendicular) line to the curve are shown at this particular point. At this point, the ray from the origin makes an incident angle of 45 degrees and the reflected ray also makes an angle of 45 degrees with respect to the normal line to the curve at this point. This is the exact forced condition that was used to generate the above differential equation – the forced condition being that the ray would experience total internal reflection.

The points satisfying the above differential equation forces the condition that all light rays emanating from the coordinate origin point will make a preferred, fixed incident angle of 45 degrees with respect to the normal line to the curve at all interception points on the reflecting curve. Thus, all these rays will undergo total internal reflection at the reflecting surface. The revolution of this two-dimensional curve about the symmetry y-axis generates the preferred three-dimensional geometrical profile of the lens shaped reflecting surface, the cross section of which is shown in Figure 3 for the Lucite-air example.

The bottom surface of this lens is shown to have a dimpled array which enhances the randomness of the exiting angles of the rays exiting from the bottom surface of the lens.

*This article was written by George A. Articolo, Ph.D., Professor of Mathematical Physics, Rutgers University (New Brunswick, NJ). For more information, contact Dr. Articolo at This email address is being protected from spambots. You need JavaScript enabled to view it..*

##### Photonics & Imaging Technology Magazine

This article first appeared in the July, 2019 issue of *Photonics & Imaging Technology* Magazine.

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