Comprehensive Notes on Ray Optics and Optical Instruments

Introduction to Light and Ray Optics

Nature has endowed the human eye, specifically the retina, with the sensitivity to detect electromagnetic waves within a small range of the electromagnetic spectrum. Electromagnetic radiation belonging to this region, with wavelengths ranging from approximately $400\,nm$ to $750\,nm$, is defined as light. Vision is the primary sense through which humans interpret the world. Common experience suggests two intuitive properties of light: it travels at an enormous speed and moves in straight lines. While the speed of light was once thought to be infinite, it is now accepted as finite and measurable. The exact value of the speed of light in a vacuum is $c = 2.99792458 \times 10^8\,m\,s^{-1}$, although for most practical calculations, it is taken as $c = 3 \times 10^8\,m\,s^{-1}$. This value represents the highest speed attainable in nature.

The apparent contradiction between light's straight-line travel and its nature as an electromagnetic wave is reconciled by the fact that the wavelength of light is extremely small compared to the size of ordinary objects, which are typically a few centimeters or larger. In these contexts, light can be modeled as traveling from one point to another along a straight line, referred to as a ray of light. A bundle of such rays is defined as a beam of light. Ray optics, or geometric optics, utilizes this ray picture to study phenomena such as reflection, refraction, and dispersion. These principles allow for the study of image formation by plane and spherical surfaces and the operation of optical instruments like the human eye, microscopes, and telescopes.

Reflection of Light by Spherical Mirrors

The laws of reflection state that the angle of incidence (the angle between the incident ray and the normal to the surface) is equal to the angle of reflection (the angle between the reflected ray and the normal). Furthermore, the incident ray, the reflected ray, and the normal at the point of incidence all lie in the same plane. These laws apply at every point on any reflecting surface, whether it is plane or curved. In the case of curved spherical mirrors, the normal is defined as the line passing through the center of curvature and the point of incidence, making it perpendicular to the tangent of the surface at that point.

Key geometric terms for spherical mirrors include the pole ($P$), which is the geometric center of the mirror, and the optical center for lenses. The center of curvature ($C$) is the center of the sphere from which the mirror was made. The line joining the pole and the center of curvature is known as the principal axis. For spherical lenses, the principal axis is the line joining the optical center with the principal focus.

Cartesian Sign Convention

To derive consistent formulas for reflection and refraction, a standard sign convention is required. The Cartesian sign convention stipulates that all distances are measured from the pole of the mirror or the optical center of the lens. Distances measured in the same direction as the incident light are considered positive, while those measured in the direction opposite to the incident light are negative. Heights measured upwards and normal to the principal axis (along the positive y-axis) are positive, while heights measured downwards are negative. This convention allows a single formula to handle various cases for both mirrors and lenses.

Focal Length of Spherical Mirrors

For paraxial rays, which are rays incident close to the pole that make small angles with the principal axis, specific convergence and divergence patterns are observed. A concave mirror causes parallel rays to converge at a point $F$ on the principal axis, known as the principal focus. Conversely, a convex mirror causes parallel rays to appear to diverge from the principal focus $F$. The distance between the focus $F$ and the pole $P$ is the focal length ($f$). If parallel rays strike the mirror at an angle to the principal axis, they converge or diverge from a point in a plane normal to the principal axis, called the focal plane.

Geometrically, it can be shown that the focal length is half the radius of curvature. Considering a ray parallel to the principal axis striking a mirror at point $M$, the angle of incidence $\theta$ results in an angle of reflection $\theta$. For small $\theta$, $\tan(\theta) \approx \frac{MD}{CD}$ and $\tan(2\theta) \approx \frac{MD}{FD}$. This simplifies to $FD \approx \frac{CD}{2}$. Since for small angles point $D$ is very close to point $P$, we conclude $f = \frac{R}{2}$. This relationship holds for both concave and convex mirrors when using paraxial rays.

The Mirror Equation and Magnification

An image is formed when rays emanating from a point on an object meet (real image) or appear to meet (virtual image) after reflection or refraction. To locate an image, at least two rays are traced: one parallel to the principal axis (which passes through the focus), one passing through the center of curvature (which retraces its path), one passing through the focus (which emerges parallel), or one incident at the pole. The mirror equation relates the object distance ($u$), image distance ($v$), and focal length ($f$) as: $\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$

Linear magnification ($m$) describes the ratio of the height of the image ($h'$) to the height of the object ($h$). Using similar triangles, magnification is also expressed in terms of distances: $m = \frac{h'}{h} = -\frac{v}{u}$

These equations are valid for all spherical mirrors, provided the Cartesian sign convention is strictly followed. For example, if a concave mirror has a focal length of $-7.5\,cm$ and an object is placed at $-10\,cm$, the image forms at $-30\,cm$ with a magnification of $-3$, indicating a real, inverted, and magnified image. If the same mirror views an object at $-5\,cm$, the image forms at $+15\,cm$ with a magnification of $+3$, indicating a virtual, erect, and magnified image behind the mirror.

Refraction and Snell's Law

Refraction is the change in direction of light as it passes obliquely from one transparent medium to another. Snell's laws of refraction state that the incident ray, the refracted ray, and the normal at the point of incidence all lie in the same plane. Furthermore, the ratio of the sine of the angle of incidence ($i$) to the sine of the angle of refraction ($r$) is a constant known as the refractive index: $n_{21} = \frac{\sin(i)}{\sin(r)}$

Here, $n_{21}$ is the refractive index of medium 2 with respect to medium 1. If n_{21} > 1, the light bends toward the normal, and medium 2 is optically denser. If n_{21} < 1, the light bends away from the normal, entering an optically rarer medium. Optical density is determined by the speed of light in the media and is not the same as mass density. For example, turpentine has a lower mass density than water but is optically denser. Refractive index depends on the pair of media and the wavelength of light. The relationship between reciprocal indices is $n_{12} = \frac{1}{n_{21}}$. In multi-media systems, $n_{32} = n_{31} \times n_{12}$.

Practical consequences of refraction include lateral shift in rectangular glass slabs, where the emergent ray is parallel to the incident ray but displaced. Another effect is apparent depth, where the bottom of a tank filled with liquid appears raised. For near-normal viewing, the relationship is: $\text{Apparent Depth} = \frac{\text{Real Depth}}{\text{Refractive Index}}$

Total Internal Reflection (TIR)

Total internal reflection occurs when light travels from an optically denser medium to an optically rarer medium. As the angle of incidence increases, the angle of refraction also increases until it reaches $90^\circ$. The specific angle of incidence that results in an angle of refraction of $90^\circ$ is called the critical angle ($i_c$). According to Snell's law: $\sin(i_c) = n_{21} = \frac{1}{n_{12}}$

If the angle of incidence exceeds the critical angle, no refraction occurs, and the light is entirely reflected back into the denser medium. Unlike reflection from mirrors, TIR involves no loss of light intensity through transmission. The critical angle for water is approximately $48.75^\circ$, for crown glass it is $41.14^\circ$, for dense flint glass it is $37.31^\circ$, and for diamond it is $24.41^\circ$. This phenomenon is used in prisms to bend light by $90^\circ$ or $180^\circ$ and in optical fibers.

Optical fibers utilize repeated TIR to transmit light signals over long distances with minimal intensity loss. They consist of a high-refractive-index core and a lower-refractive-index cladding. Light entering the core at a suitable angle undergoes successive internal reflections. Modern silica glass fibers can transmit more than 95% of light over a length of $1\,km$. These are used in telecommunications and medical instruments like endoscopes to examine internal organs.

Refraction at Spherical Surfaces and Lenses

Refraction at a single spherical surface separating two media of indices $n_1$ and $n_2$ is governed by the relation: $\frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R}$

A thin lens is a transparent medium bounded by two surfaces, at least one of which is spherical. By applying the single-surface refraction formula to both sides of a lens, we derive the Lens Maker's Formula: $\frac{1}{f} = \left(\frac{n_2}{n_1} - 1\right) \left(\frac{1}{R_1} - \frac{1}{R_2}\right)$

Combined with the thin lens formula: $\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$

Magnification for a lens is given by $m = \frac{v}{u}$. For a convex lens, $f$ is positive (converging), and for a concave lens, $f$ is negative (diverging). If a lens is placed in a liquid with a refractive index equal to its own, it becomes invisible and its focal length becomes infinite, acting as a plane sheet of glass.

Power and Combination of Lenses

The power ($P$) of a lens measures its ability to converge or diverge light and is defined as the reciprocal of its focal length in meters: $P = \frac{1}{f}$

The SI unit for power is the dioptre ($D$), where $1\,D = 1\,m^{-1}$. Converging lenses have positive power, while diverging lenses have negative power. For example, a lens with $P = +2.5\,D$ has a focal length of $+40\,cm$.

When multiple thin lenses are placed in contact, the total power is the algebraic sum of individual powers: $P = P_1 + P_2 + P_3 + \dots$ $\frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2} + \frac{1}{f_3} + \dots$

The total magnification of the combination is the product of individual magnifications: $m = m_1 \times m_2 \times m_3 \times \dots$

Such combinations are essential in high-quality optical instruments to improve image sharpness and minimize optical aberrations like chromatic distortion.

Refraction Through a Prism

In a triangular glass prism, a ray of light undergoes two refractions, once at each face. The angle between the incident ray and the emergent ray is called the angle of deviation ($\delta$). It depends on the angle of incidence ($i$), the prism's angle ($A$), and the material's refractive index. The geometric relations are: $r_1 + r_2 = A$ $\delta = i + e - A$

There exists a minimum deviation ($D_m$) where the refracted ray inside the prism is parallel to the base. At this point, $i = e$ and $r_1 = r_2 = \frac{A}{2}$. The refractive index of the prism material is then given by: $n_{21} = \frac{\sin\left(\frac{A + D_m}{2}\right)}{\sin\left(\frac{A}{2}\right)}$

For thin prisms (small $A$), the deviation is approximately $\delta = (n_{21} - 1)A$. Prisms are fundamental in spectroscopic analysis and for inverting images in instruments.

Optical Instruments: The Microscope

A simple microscope consists of a single converging lens of small focal length. It produces a virtual, erect, and magnified image. Maximum magnification occurs when the image is at the near point ($D \approx 25\,cm$): $m = 1 + \frac{D}{f}$

If the image is at infinity (viewed by a relaxed eye), the angular magnification is: $m = \frac{D}{f}$

A compound microscope uses two lenses: the objective (near the object) and the eyepiece (near the eye). The objective forms a real, inverted, magnified image, which then acts as an object for the eyepiece. The total magnification is: $m = m_o \times m_e \approx \frac{L}{f_o} \times \frac{D}{f_e}$

Here, $L$ is the tube length, the distance between the second focal point of the objective and the first focal point of the eyepiece. Large magnifications require both lenses to have very short focal lengths.

Optical Instruments: The Telescope

Telescopes provide angular magnification for distant objects. They consist of an objective with a large focal length and aperture, and an eyepiece with a small focal length. The magnifying power ($m$) is the ratio of the angle subtended by the image to the angle subtended by the object: $m = \frac{f_o}{f_e}$

In refracting telescopes, the tube length is $f_o + f_e$. Large refractors are limited by lens weight and chromatic aberration. Modern astronomical telescopes are reflecting telescopes using concave mirrors as objectives. Reflecting telescopes, such as the Cassegrain design, use a primary concave mirror and a secondary convex mirror. They eliminate chromatic aberration and offer better mechanical support for large diameters. Examples include the Keck telescopes in Hawaii ($10\,m$ diameter) and the Vainu Bappu Telescope in Kavalur, India ($2.34\,m$ diameter).