Geometric vs physical optics

• Light is a wave, so it spreads out as it propagates.
• Geometric optics ignores this spreading and assume light travels in straight lines as light rays. This is what we study this chapter.
• Physical optics takes into account the wave nature of light. This is studied in later chapters.

Reflection and refraction

• Some of the light gets reflected back into the first medium (reflection).
• Some may be (but not always) transmitted to the second medium (refraction).

Zátonyi Sándor (ifj.) / CC BY-SA

The law of reflection is:

• Normal is a line perpendicular to the boundary between the two media.
• Both the angle of incidence and angle of reflection are measured with respect to the normal.
• The complete law has a little more details but this simplified version is good enough for us.

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Image produced by a flat mirror

• Follow any light rays that enter the eyes backward. Wherever the rays meet is the location of the image, where we perceive as the source of light.
• Draw any light rays from the object, use the law of reflection to draw the reflected rays. The reflected rays all lead backward and cross the location of the image. You can try this yourself on a piece of paper.
• The image is as far behind the mirror as the object is in front (object distance = image distance).
• That is why you see a copy of yourself behind the mirror when you brush your teeth in the morning. It is an optical illusion.

The index of refraction (or refractive index) of a material $X$ is given by:

• The index of refraction is dimensionless (i.e. no units).
• $c_X$: speed of light in medium $X$.
• $c$: speed of light in vacuum.
• $n \geq 1$, equality only when $X$ is vacuum.
• Large $n_X$ means small $c_X$ and slower light speed in $X$.
• $n_{air} \approx 1$, which you should remember. The refractive index for all other materials will be given to you in the exam.

When light goes from one medium into another, the ray gets refracted according to Snell's law of refraction:

From the law of refraction we can deduce the following cases:

• If $n_1 \lt n_2 \Rightarrow \theta_1 \gt \theta_2$: ray bends toward the normal.
• If $n_1 \gt n_2 \Rightarrow \theta_1 \lt \theta_2$: ray bends away from the normal.

Lucas Vieira / Public domain

Peo / CC BY-SA

The refractive index of a material has a slight dependence on the wavelength (or color in the case of visible light). Therefore different wavelengths refract slightly differently.

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Josell7 / CC BY-SA

• When light enters from $n_1$ to $n_2$, some light is reflected back while some is refracted into the second medium.
• However, it is possible for 100% of the light to get reflected, leaving no refracted ray. This is called total internal reflection.
• Mathematically, total internal reflection happens when the Snell's law of refraction cannot be satisfied, so there is no solution for the angle of refraction.
• Total internal refection is only possible when light enters from large $n$ to small $n$, i.e. if $n_1 \gt n_2$ so $\theta_1 \lt \theta_2$.
• Use the earlier simulation to explore total internal reflection.

Critical angle

• The critical angle ($\theta_{critical}$) is the angle of incidence when the angle of refraction is $90^\circ$.
• Total internal reflection occurs when $\theta_{incidence} \gt \theta_{critical}$. No refracted ray.
• The critical angle is always found on the side with the larger $n$.

To find the critical angle, set $\theta_1 = \theta_{critcal}$ and $\theta_2 = 90^\circ$: $$ \begin{eqnarray} n_1 \sin \theta_{critical} &=& n_2 \sin 90^\circ \\ &=& n_2 \\ \Rightarrow \sin \theta_{critical} &=& \frac{n_2}{n_1} \\ \Rightarrow \theta_{critical} &=& \sin^{-1} \frac{n_2}{n_1} \end{eqnarray} $$ $\theta_{critical}$ only exists if $n_1 \gt n_2$. There is no total internal reflection if $n_1 \lt n_2$.

Jean-Marc Kuffer / CC BY

DPoptics / CC BY-SA

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Optical fiber

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