Comprehensive Study Notes on the Refraction of Light and Lens Theory

Fundamentals of Refraction and the Refractive Index

Refraction is defined as the bending of light at an interface between two media of different optical densities. It can also be characterized as the change in the speed or the direction of light as it travels from one medium to another. When light moves from an optically less dense medium into a denser medium, it is refracted toward the normal at the point of incidence. Conversely, light traveling from a denser medium to a less dense medium bends away from the normal.

The study of refraction is governed by two primary laws. The first law states that the incident ray, the normal, and the refracted ray at the point of incidence all lie in the same plane. The second law, known as Snell’s law, states that the ratio of the sine of the angle of incidence to the sine of the angle of refraction is constant for a given pair of media. Mathematically, this constant is known as the refractive index of the two media. The refractive index ($n$) can also be defined as the ratio of the speed of light in a vacuum ($c = 3.0 \times 10^8\,ms^{-1}$) to the speed of light in a specific medium ($v$), represented by the equation $n = \frac{c}{v}$.

The refractive index for a vacuum is exactly $1$, while for other media it is slightly higher, such as $1.33$ for water and approximately $1.5$ for glass. The Principle of Reversibility of light states that the paths of light rays are reversible, meaning a ray can travel from medium $1$ to $2$ and from $2$ to $1$ along the identical geometric path. For light moving through multiple media, the relationship between indices is represented as $_1n_2 \times _2n_1 = 1$. For a ray passing through a series of media (1, 2, 3), the index between media 2 and 3 can be expressed as $_2n_3 = _2n_1 \times _1n_3 = \frac{n_3}{n_2}$. The general relation for light passing through multiple plane boundaries is expressed as $n\sin(i) = \text{constant}$.

Sidewise Displacement and Total Internal Reflection

When light travels through a parallel-sided medium like a glass block, it suffers lateral displacement, which is a sidewise shift in its original direction. For a ray incident at angle $i$ on a block of thickness $t$ and refracted at angle $r$, the lateral displacement ($d$) is derived as $d = \frac{t\sin(i-r)}{\cos(r)}$. The horizontal displacement of the incident ray across the top surface is given by $t\tan(r)$.

The critical angle ($c$) is the specific angle of incidence in a denser medium that results in an angle of refraction of $90^0$ in the adjoining less dense medium. If the angle of incidence exceeds this critical point, all incident light energy is reflected back into the denser medium, a phenomenon called total internal reflection. The critical angle is calculated using the formula $\sin(c) = \frac{n_2}{n_1}$, where $n_1$ is the index of the denser medium and $n_2$ is the index of the less dense medium. If the less dense medium is air ($n = 1$), then $\sin(c) = \frac{1}{n}$.

Total internal reflection requires two conditions: light must travel from an optically denser to a less dense medium, and the angle of incidence must be greater than the critical angle. This principle explains various phenomena and technological applications, including the formation of mirages on hot days where air layers near the ground are less dense, the formation of rainbows, and the transmission of light in optical fibers. In medical practice, optical fibers are used in endoscopes to view internal organs, and in telecommunications, they carry laser-light signals for telephone and TV systems. It also allows sky radio waves to travel from a transmitting station to a receiver by reflecting off ionized gas in the atmosphere.

Methods for Measuring Refractive Index

The air-cell method is an efficient way to estimate the refractive index of a liquid. An air cell is immersed in the liquid under test, and monochromatic light is directed through it. The cell is rotated until light is cut off at angular positions $\theta_1$ and $\theta_2$. The refractive index is calculated from $n = \frac{1}{\sin(\theta)}$ where $\theta = \frac{\theta_1 + \theta_2}{2}$. Monochromatic light is necessary to ensure a sharp extinction of light by avoiding dispersion. Theoretical analysis using the relation $n\sin(i) = \text{constant}$ at the liquid-glass-air interfaces further confirms this relationship.

The refractive index of a glass block or a liquid can also be determined via the apparent depth method. When an object is viewed normally through a medium of thickness $t$, it appears to be at a shallower depth. The refractive index is defined as the ratio of the real depth to the apparent depth. The apparent displacement ($d$) of the image is given by $d = t(1 - \frac{1}{n})$. For multiple layers of different materials, the total displacement is the sum of the separate displacements. A traveling microscope is typically used in these experiments to focus on particles at the bottom of a container, the displaced image, and the top surface of the material to measure these heights accurately.

A concave mirror can also be used to find the refractive index of a small quantity of liquid. By placing a liquid in the mirror and moving an object pin until it coincides with its image at point $I$, the index of the liquid is determined as $n_l = \frac{r}{IP}$, where $r$ is the radius of curvature of the mirror and $IP$ is the distance from the pin to the mirror pole. If the liquid depth ($d$) is significant, the formula becomes $n_l = \frac{r-d}{MI}$, where $MI$ is the height of the pin above the liquid surface.

Prism Optics and Dispersion

A prism deviates light by an angle $D$, defined as the angle between the incident and emergent rays. For a prism with refracting angle $A$, the total deviation is $D = (i_1 + i_2) - A$, and the relationship between internal angles is $r_1 + r_2 = A$. As the angle of incidence increases, the deviation decreases to a minimum value ($D_{\text{min}}$) and then increases again. At minimum deviation, light passes through the prism symmetrically, meaning $i_1 = i_2 = i$ and $r_1 = r_2 = r = \frac{A}{2}$. The refractive index of the prism material is then $n = \frac{\sin(\frac{A + D_{\text{min}}}{2})}{\sin(\frac{A}{2})}$.

For small-angle prisms, where angles are measured in radians, the deviation is approximately independent of the angle of incidence and is given by $D = (n - 1)A$. This leads to the phenomenon of dispersion, where white light is separated into its component colors (the spectrum: red, orange, yellow, green, blue, indigo, violet) because different colors travel at different speeds and are refracted differently. Violet is the most refracted and red is the least refracted. The angular separation (or angular dispersion) between two wavelengths is given by $\phi = (n_b - n_r)A$.

The grazing property occurs if a ray incident or emergent is at $90^0$ to the normal. This leads to the limiting angle of a prism, which is the maximum refracting angle ($A = 2c$) for which an emergent ray can still graze the second surface. A spectrometer is a precision instrument consisting of a collimator, a turntable, and a telescope used to measure these angles. It requires three adjustments: setting the collimator for parallel rays, leveling the turntable, and focusing the telescope for the cross-wires.

Lens Theory and Common Formulas

A lens is a piece of glass bounded by spherical surfaces, classified as convex (converging) or concave (diverging). Key definitions include the centers and radii of curvature, the principal axis, and the optical center. The principal focus ($F$) is the point where parallel rays converge (for convex) or appear to diverge from (for concave), and the focal length ($f$) is the distance from the optical center to this focus. Images formed by convex lenses vary based on object distance: they can be real or virtual, erect or inverted, and magnified or diminished. Concave lenses always produce virtual, erect, and diminished images.

The thin lens formula is $\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$, where $u$ is the object distance and $v$ is the image distance. The sign convention adopted is that distances of real objects and images are positive, while virtual ones are negative. Consequently, a convex lens has a positive focal length, and a concave lens has a negative focal length. The power of a lens is the reciprocal of the focal length in meters, measured in dioptres ($D = \frac{1}{f}$). Linear magnification ($m$) is the ratio $\frac{v}{u}$. For a convex lens, the minimum distance between a real object and its real image is $4f$.

Newton’s formula for lenses states $f^2 = xy$, where $x$ and $y$ are the distances of the object and image from their respective focal points. If two thin lenses are in contact, their combined focal length ($F$) is given by $\frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2}$. The lens maker's formula relates the focal length to the refractive index and radii of curvature: $\frac{1}{f} = (n - 1)(\frac{1}{r_1} + \frac{1}{r_2})$. For biconvex lenses, radii are typically positive, while for biconcave or meniscus lenses, the signs vary based on curvature direction.

Lens Aberrations and Correction

Lenses suffer from two main types of defects: chromatic aberration and spherical aberration. Chromatic aberration occurs because the refractive index of glass varies with wavelength, causing different colors of white light to focus at different points. This can be corrected using an achromatic doublet, which combines a convex and a concave lens of different materials (like crown and flint glass) so that the dispersion of one cancels the other. Alternatively, placing the eye very close to the lens can minimize the perception of color separation.

Spherical aberration happens when a lens fails to focus all rays (especially marginal versus paraxial rays) to the same point, resulting in a blurred image. This is more pronounced in lenses with large apertures. It can be minimized by using aspheric surfaces, utilizing lenses with small apertures, or employing stoppers to block marginal rays. These adjustments allow all light to align more accurately at a single focal point, though stoppers may reduce image brightness by limiting the amount of light entering the system.

Questions & Discussion

Various exercises explore the practical application of these principles. One scenario involves calculating the time light takes to pass through a block: for a block of thickness $t$ and refractive index $n$ with refraction angle $\theta$, the time is $T = \frac{nt\sec(\theta)}{c}$. In another case, the shift of an object viewed through multiple media (liquid, glass, water) is calculated by summing individual displacements $d = t(1 - \frac{1}{n})$. The displacement method for finding the focal length of a lens in a tube involves fixed object and screen positions and moving the lens between two points of focus; the focal length is then $f = \frac{L^2 - d^2}{4L}$, and the object height can be determined using $h = \sqrt{h_1 h_2}$.

Additional discussions focus on why agriculturalists or doctors might use specific lenses, such as convex lenses for long-sightedness and concave for short-sightedness. The formation of spectra in raindrops (rainbows) involves both dispersion and total internal reflection. In reflecting telescopes, spherical mirrors are often used instead of lenses to eliminate chromatic aberration entirely. Finally, scientific observations such as why a fish appears larger in water are explained by the surface ripples acting as magnifying lenses when the subject is within their focal length.