— P R O B L E M S —

Light, transverse electromagnetic wave, propagates without any medium. Speed of light was measured by Ole Rømer in the seventeenth century using the following mechanism.

Romer observed the emergence from (figure b) and immersion into (figure a) eclipse of the moon Io due to the planet Jupiter. During one period of Io ($\sim 42$ hours), Earth moved from $A$ to $B$ or, 6 months later, from $A'$ to $B'$. From the differences in light travel time at these points, he measured the speed of light in vacuum to be around $2\times 10^8$ m/s which is very close to the current value

$$c=299792458 \text{ m/s} \approx 3\times 10^8 \text{ m/s}.$$

Light is electromagnetic (EM) wave. Any accelerated charged particle emits EM waves. The mechanism can be seen in this animation.

Speed of light is different in different material. If the index of refraction of a material is $n$, the speed of light in that material is $v$ and the speed of light in vacuum is $c$, then

$$ n = \frac{c}{v} $$

where $n\ge 1$. The index itself varies slightly with wavelength. Index of refraction of Hydrogen is 1.000139 and for diamond is 2.419. For most gases $n\sim 1$.

The path of light through an unobstructed medium can be modelled as a straight line called ray. In ray optics, we focus on the particle-like property of light by assuming that a photon is traveling a bullet along a straight path.

We see through reflections of light from an object.

The law of reflection states that the angle of incidence ($\theta_i$) is always equal to the angle of reflection ($\theta_r$).

Images can be formed by mirrors because of this single angle of reflection. Images are formed behind a mirror, so a mirror hanging on a wall in a room will make the room seem bigger.

A light ray that strikes two mutually perpendicular reflecting surfaces is reflected back exactly parallel to the direction from which it came. This principle is used to create a corner reflector or retoreflector. Apollo astronauts placed a corner reflector on the surface of the moon.

Light rays change directions when passing through substances of different refractive indices. This is called refraction. The change in direction is related to the change in speed of light; $v=c/n$.

If a light ray passes from Medium 1 (refractive index $n_1$) to Medium 2 ($n_2$) or vice versa, the law of refraction (or Snell’s law) states that

$$ n_1 \sin\theta_1 = n_2 \sin\theta_2 $$

where $\theta_1$ and $\theta_2$ are the incident and refracted angles, respectively, or vice versa. Dutch physicist Snell derived this law without knowing about the change in speed of light.

We can get reflection from refraction if the incident angle is large enough.

If incident angle $\theta_1$ is such that refracted angle $\theta_2>90^\circ$, the incident ray is completely reflected. This is called total internal reflection. The incident angle at which this happens is called the critical angle $\theta_c$. Putting $\theta_2=90^\circ$ in Snell’s law gives us the condition for the critical angle:

$$ n_1 \sin\theta_1 = n_2 \Rightarrow \theta_c = \sin^{-1}\frac{n_2}{n_1} $$

when light rays enter from Medium 1 to Medium 2 and $n_1>n_2$. So $n_2/n_1<1$ and $\theta_c<90^\circ$.

Total internal reflection is used in fiber optics. Optic fibers are thin, so light entering at one end are continuously totally reflected. Optic fibers are used in communication.

Index of refraction $n$ depends on wavelength. So light rays transitioning from one medium to another get dispersed into different wavelengths.

If monochromatic light (only one wavelength $\lambda$) enters a prism (figure a), it will exit the prism at a specific angle after two refractions. If white light (many wavelengths) enters a prism, it is split into different color, i. e. wavelengths. As $n$ depends on $\lambda$, angles of refraction also depend on $\lambda$ because of Snell’s law.

Index of refraction of glass is 1.662 for red light (660 nm) but 1.684 for blue light (470 nm). So blue light will exhibit higher angles of refraction.

Rainbow is created when sunlight entering water droplets disperse after two refractions and a total internal reflection.

Rainbows are seen as arc because of the required geometric relationship of the sun, the rainbow and the observer. Secondary inverted rainbows are created if light is reflected twice; i. e. if there are 2 refractions and 2 total internal reflections.

Light is an electromagnetic (EM) wave as mentioned above. EM waves are transverse. Normally electric ($\mathbf{E}$) and magnetic ($\mathbf{B}$) fields oscillate in random directions perpendicular to the direction of propagation. In this case the light is unpolarized.

But if $\mathbf{E}$ and $\mathbf{B}$ oscillate in preferred directions, for example $\mathbf{E}$ oscillates vertically and $\mathbf{B}$ horizontally in the figure above, then the wave is polarized.

Unpolarized light can become polarized after passing through a polarizing filter as shown above. The filter only allows the components of $\mathbf{E}$ that are parallel to its own axis.

Intensity $I$ of a wave is proportional to its amplitude squared because energy carried by a wave $E\propto A^2$. So $I$ is also proportional to amplitude squared and, hence,

$$ I = I_0 \cos^2\theta $$

which is called Malus’s law after its originator. We can get partial polarization via reflection.

Unpolarized light has equal amount of vertically and horizontally polarized components. After interaction with a surface, the horizontal components are refracted more and vertical components reflected more. So the reflected light is partially vertically polarized and the refracted light is partially horizontally polarized.

Reflected light is completely polarized at an angle of reflection

$$ \theta_b = \tan^{-1} \frac{n_2}{n_1} $$

which is also called Brewster’a angle. The laws is also called Brewster’s law. Here $n_1$ is the index for the medium and $n_2$ for the reflecting surface.

Unpolarized light can be polarized via scattering as well. Transverse EM waves force the electrons in gas atoms to oscillate in their directions. A vertically oscillating electron radiates in the horizontal direction. So an observer at 90 degrees with the direction of propagation sees horizontally polarized light. So light scattered by air is partially polarized.

The field of optics where light is considered a ray is called geometric optics. Ray tracing sets the rules of the game here.

Reflected images are formed behind a plane mirror. Trace the rays to find out the location of the image and you will see that the object distance $d_o$ (from the mirror) and the image distance $d_i$ (from mirror) are related as

$$ d_o = -d_i. $$

The image in a plane mirror has the same size as the object, is upright, and is the same distance behind the mirror as the object is in front of the mirror. A curved mirror, on the other hand, can form images that may be larger or smaller than the object and may form either in front of the mirror or behind it.

The image in a plane mirror has the same size as the object, is upright, and is the same distance behind the mirror as the object is in front of the mirror. A curved mirror, on the other hand, can form images that may be larger or smaller than the object and may form either in front of the mirror or behind it. Curved mirrors are therefore used in many different optical devices.

Mirrors are symmetric with respect to the optical axis or principal axis that passes through the center of curvature and the vertex. Convex mirrors have reflecting surface on the outer side and concave mirrors on the inner side.

Rays parallel to the optical axis of a parabolic mirror meet at a single point after reflection from the mirror surface because of the law of reflection. This point is called the focal point of a mirror. If a spherical mirror is large compared to its radius of curvature, rays will not meet at a single point causing spherical aberration. However, if the spherical mirror is small compared to its radius of curvature, something similar to the parabolic mirror will happen.

The distance along the optical axis from the focal point to the mirror (or vertex) is called the focal length of a mirror.

In this mirror, C is center of curvature. So radius of curvature $R=CF+FP$. Triangle $CFX$ is an isoceles, so $CF=FX$. And if angle $\theta$ is small, $FX=FP$. So $CF=FP=f$ where $f$ is the focal length. Therefore,

$$ f = \frac{R}{2}. $$

Now let us try to understand the mirror equation and magnification.

The height of the object is $h_o$ and that of the image is $h_i$. Similarly $d_o$ and $d_i$ are distances of the object and the image from the vertex.

The figure shows, $\tan\theta=h_o/d_o$ and $\tan\theta'=-\tan\theta=h_i/d_i$ which means

$$ m = \frac{h_i}{h_o} = - \frac{d_i}{d_o} $$

where $m$ is the linear magnification or just magnification. Following the train of thought it can be proven that

$$ \frac{1}{d_o} + \frac{1}{d_i} = \frac{1}{f} $$

which is the mirror equation. It reduces to the equation for plane mirror if $f=\infty$. The focal point of a plane mirror ($d_o=-d_i$) is indeed at infinity.

A rod submerged in water appears bent because of refraction. This is a good starting point for understanding images formed by refraction.

Actual object depth (the part submerged in water) $h_o$ appears to be smaller. The apparent depth

$$ h_i = \left(\frac{n_2}{n_1}\right) h_o. $$

A fish typically appears 3/4 of the real depth when seen from above.

The figure shows images formed by refraction at a convex spherical surface with center of curvature $C$. Light enters from medium with index of refraction $n_1$ to a medium with index $n_2$ where $n_2>n_1$. Beginning from Snell’s law it can be proven that

$$ \frac{n_1}{d_o} + \frac{n_2}{d_i} = \frac{n_2-n_1}{R} $$

where $R$ is the radius of curvature of the convex surface. Two focal points can be defined from this equation called the first or object focus and the second or image focus.

If an object is placed at first focus $F_1$, image will be formed at infinity. Putting $d_i=\infty$ and $d_o=f$ in the previous equation we get

$$ f_1 = \frac{n_1R}{n_2-n_1}. $$

On the other hand if the object is at infinity the image will be formed at the second focus. Putting $d_o=\infty$ and $d_i=f$ in the same equation we get

$$ f_2 = \frac{n_2R}{n_2-n_1}. $$

There are many different types of lenses broadly classified into two categories: converging and diverging.

In both cases, the refracted rays converge at the focal point. A lens has two refracting surfaces that form two different images. The image formed by one surface works as the object for the other surface. If the lens is very thin compared to the distance to the first image, it is considered a thin lens.

The thin lens in this figure has two convex refracting surfaces with radii or curvature $R_1$ and $R_2$. The lens medium has refractive index $n_2$ and the surrounding medium has index $n_1$. Using the equation shown above for refractive image formation, it can be shown that

$$ \frac{1}{d_o} + \frac{1}{d_i} = \frac{1}{f} $$

which is called the lens equation; exactly the same as the mirror equation. This leads to the lensmaker’s equation:

$$ \frac{1}{f} = \left(\frac{n_2}{n_1}-1\right) \left(\frac{1}{R_1}-\frac{1}{R_2}\right). $$

If the surrounding medium is air, $n_1\approx 1$ and $n_2\equiv n$ and the equation becomes

$$ \frac{1}{f} = (n-1) \left(\frac{1}{R_1}-\frac{1}{R_2}\right). $$

The figure below show the image distance as a function of object distance for a converging lens based on the above equation. The focal length can be varied.

The following sign conventions need to be maintained for lenses

1. Image distance $d_i$ is positive if the image is on the side opposite the object (real image), otherwise, $d_i$ is negative (virtual image).
2. $f$ is positive for a converging lens and negative for a diverging lens.
3. $R$ is positive for a surface convex toward the object, and negative for a surface concave toward object.