Physics

Chapter 1: The Nature of Light

Imron Rosyadi

Chapter 1: The Nature of Light

Note

Learning Objectives

By the end of this chapter, you will be able to:

• Understand how light propagates and interacts with different media.
• Explain the principles of reflection, refraction, and total internal reflection.
• Describe the phenomena of dispersion, Huygens’s principle, and polarization.

1.1 The Propagation of Light

• Light travels at a finite speed, c, in a vacuum.
• This speed is one of physics’ fundamental constants: \(c = 2.99792458 \times 10^8 \text{ m/s} \approx 3.00 \times 10^8 \text{ m/s}\).
• Light’s speed changes in different materials.

In this section, we’ll explore how light travels and how its speed is measured.

We’ll also introduce the concept of the index of refraction.

Early Measurements of Light Speed

Ole Roemer (1675)

• Studied Io, a moon of Jupiter.
• Noticed fluctuations in Io’s eclipse period depending on Earth’s position.
• Calculated \(c \approx 2.0 \times 10^8 \text{ m/s}\) (33% below accepted value).

Figure 1.2: Roemer’s method to determine the speed of light.

Roemer’s observation was groundbreaking. The changing distance between Earth and Jupiter caused delays in observing Io’s eclipses, directly related to the finite speed of light. Although his value was approximate, it was the first quantitative estimate.

Terrestrial Measurements

Armand Fizeau (1849)

• First successful terrestrial measurement.
• Used a rapidly rotating toothed wheel and a mirror 8 km away.
• Measured \(c \approx 3.15 \times 10^8 \text{ m/s}\) (5% too high).

Figure 1.3: Fizeau’s method for measuring the speed of light.

Fizeau’s method was ingenious, using mechanical means to chop light into pulses and measure their travel time over a known distance. This was a significant step as it didn’t rely on astronomical observations.

Modern Measurements

• Jean Bernard Léon Foucault (1862): Modified Fizeau’s method with a rotating mirror.
  • Measured \(c \approx 2.98 \times 10^8 \text{ m/s}\) (within 0.6% of accepted value).
• Albert Michelson (1878-1926): Refined Foucault’s technique.
  • Achieved (2.99796 ± 4) \(\times 10^8 \text{ m/s}\).
• Present Day: The speed of light in a vacuum is defined as \[ c = 2.99792458 \times 10^8 \text{ m/s} \] Approximately \(3.00 \times 10^8 \text{ m/s}\) for typical calculations.

Foucault’s and Michelson’s work greatly improved the precision of light speed measurements, demonstrating the incredible consistency of this fundamental constant. The modern definition of ‘c’ is now exact, and the meter is defined based on it.

Speed of Light in Matter: Index of Refraction

• Light travels slower in matter than in a vacuum.
• This reduced speed is described by the index of refraction (n). \[ n = \frac{c}{v} \] where v is the speed of light in the material.
• Since \(v \le c\), then \(n \ge 1\).
• n varies slightly with the wavelength of light.

The index of refraction is a crucial property of a material, telling us how much the speed of light is reduced within it. This change in speed is responsible for many optical phenomena, including refraction itself.

Index of Refraction in Various Media

Medium	n
Air (0°C, 1 atm)	1.000293
Water, fresh (20°C)	1.333
Glass, crown (20°C)	1.52
Diamond (20°C)	2.419

Table 1.1: Representative indices of refraction for light with wavelength 589 nm.

Tip

For gases, n is very close to 1.0, meaning light travels almost as fast as in a vacuum.

Notice how dramatically the index of refraction can vary, especially for materials like diamond. This difference in ‘n’ is what makes diamonds sparkle. The slight variation with wavelength, though not always significant, is key to understanding dispersion.

Example: Speed of Light in Zircon

Question: Calculate the speed of light in zircon, which has an index of refraction of 1.923.

Strategy:

Use the definition of the index of refraction: \(n = c/v\).

Rearrange to solve for \(v\): \(v = c/n\).

Solution: Given \(c = 3.00 \times 10^8 \text{ m/s}\) and \(n = 1.923\). \[ v = \frac{3.00 \times 10^8 \text{ m/s}}{1.923} \approx 1.56 \times 10^8 \text{ m/s} \]

Significance: The speed of light in zircon is about half its speed in a vacuum.

This calculation demonstrates the direct relationship between the index of refraction and the speed of light in a medium. A higher index means a slower speed.

The Ray Model of Light

• Light travels in straight lines called rays.
• This model is valid when light interacts with objects much larger than its wavelength (e.g., visible light and objects > 1 micron).
• Geometric optics studies light propagation using the ray model.
• Light changes direction when:
  1. Reflected from a surface (e.g., a mirror).
  2. Refracted when passing from one material to another (e.g., air to glass).

The ray model simplifies the study of light by treating it as straight lines, which makes many optical phenomena understandable through geometry. This is applicable in everyday situations where objects are much larger than light’s wavelength.

How Light Travels

Figure 1.4: Three methods for light to travel from a source to another location.

1. Direct travel through empty space, like sunlight to Earth.
2. Travel through media, like light through a window to an observer.
3. Reflection from an object, like light off a mirror. In all these cases, the ray model applies, showing light interacting with objects much larger than its wavelength.

1.2 The Law of Reflection

• When light strikes a surface, it bounces off according to the law of reflection.
• The angle of reflection (\(\theta_r\)) equals the angle of incidence (\(\theta_i\)). \[ \theta_r = \theta_i \]
• Angles are measured relative to the normal (perpendicular line) to the surface.

The law of reflection is fundamental to understanding how mirrors work and how light interacts with surfaces. It’s a simple yet powerful rule that governs phenomena from seeing your image in a mirror to how telescopes collect light.

Specular vs. Diffuse Reflection

Smooth Surfaces (Specular Reflection)

• Reflect light at specific angles.
• Form clear images (e.g., mirrors, still water).

Figure 1.5: Specular reflection from a smooth surface.

Rough Surfaces (Diffuse Reflection)

• Scatter light in many directions.
• Allow objects to be seen from any angle (e.g., paper, clothing).

Figure 1.6: Diffuse reflection from a rough surface.

The nature of the surface dictates whether reflection is specular or diffuse. Specular reflection is like a well-organized march, while diffuse reflection is like a crowd dispersing in all directions. Both are governed by the law of reflection at the microscopic level, but the macroscopic appearance differs due to the surface’s texture relative to the light’s wavelength.

Images from Mirrors

• Mirror images appear to be behind the mirror.
• The image distance behind the mirror is equal to the object distance in front.
• These images are not optical illusions; they can be captured by cameras.
• This phenomenon makes rooms appear larger.

Figure 1.8: Your image in a mirror appears behind it, at the same distance as you are in front.

The formation of images by mirrors is a core concept in geometric optics. The light rays obey the law of reflection, and our brains interpret these reflected rays as originating from a point behind the mirror. This is how virtual images are formed.

Corner Reflectors (Retroreflectors)

• An object with two (or three) mutually perpendicular reflecting surfaces.
• Reflects incoming light rays back exactly parallel to their original direction.
• Independent of the angle of incidence.

Figure 1.9: A light ray striking two perpendicular mirrors is reflected back parallel to its incident direction.

Corner reflectors are incredibly useful because they return light directly to its source. This property is due to successive reflections off perpendicular surfaces.

Applications of Corner Reflectors

• Bicycle/Car Reflectors: Enhance visibility for drivers.
• Apollo Moon Reflectors: Used by astronauts to measure the Earth-Moon distance with lasers.
• Radar Reflectors: Make small boats visible to radar systems, preventing collisions.

Figure 1.10: (a) Astronaut placing a corner reflector on the Moon. (b) Bicycle safety reflectors.

The versatility of corner reflectors is evident in their wide range of applications, from everyday safety items to advanced scientific instruments. Their ability to return light to the source makes them highly effective in situations where visibility or precise distance measurement is critical.

1.3 Refraction

• Refraction is the bending of light as it passes from one medium to another.
• Caused by a change in the speed of light (\(v = c/n\)).
• Responsible for phenomena like lenses, optical fibers, and apparent distortions in water.

Figure 1.12: Fish appearing in two places due to refraction.

Refraction is one of the most common optical phenomena we encounter. Think about a straw in a glass of water appearing bent, or how eyeglasses correct vision. The underlying cause is the change in the speed of light as it crosses the boundary between different materials, which in turn changes its direction.

Direction of Refraction

• When light enters a medium with a higher index of refraction (slowing down), it bends towards the normal.
• When light enters a medium with a lower index of refraction (speeding up), it bends away from the normal.
• The path of light is reversible.

Figure 1.13: Light bending (a) towards the normal (\(n_2 > n_1\)) and (b) away from the normal (\(n_2 < n_1\)).

It’s important to remember these rules for bending. Imagine light as a car. If one wheel hits a patch of mud (slower medium) before the other, the car will turn. The same principle applies to light waves entering a new medium at an angle.

Snell’s Law (Law of Refraction)

• Describes the quantitative relationship between angles and indices of refraction. \[ n_1 \sin \theta_1 = n_2 \sin \theta_2 \]
  • \(n_1, n_2\): indices of refraction for medium 1 and 2.
  • \(\theta_1\): incident angle (in medium 1).
  • \(\theta_2\): refracted angle (in medium 2).

Note

Discovered by Willebrord Snell in 1621, and earlier by Ibn Sahl in 984.

Snell’s Law is a cornerstone of geometric optics, allowing us to precisely calculate how much light will bend when it passes between different media. It connects the angles of incidence and refraction with the properties of the materials themselves.

Example: Determining the Index of Refraction

Question: Light goes from air (medium 1) into an unknown medium (medium 2). If the incident angle is \(30.0^\circ\) and the refracted angle is \(22.0^\circ\), find \(n_2\).

Strategy:

Use Snell’s law: \(n_1 \sin \theta_1 = n_2 \sin \theta_2\).

Given \(n_1 = 1.00\) (air), \(\theta_1 = 30.0^\circ\), \(\theta_2 = 22.0^\circ\).

Solve for \(n_2\).

Solution:

\[ n_2 = \frac{n_1 \sin \theta_1}{\sin \theta_2} = \frac{(1.00) \sin 30.0^\circ}{\sin 22.0^\circ} = \frac{0.500}{0.375} \approx 1.33 \]

Significance:

This value, \(n_2 = 1.33\), is the index of refraction for water.

This example shows how Snell’s law can be used to identify an unknown material by measuring the bending of light. It highlights the practical application of the law.

Example: A Larger Change in Direction

Question: If light goes from air to diamond with an incident angle of \(30.0^\circ\), calculate the refracted angle (\(\theta_2\)).

Strategy:

Use Snell’s law: \(n_1 \sin \theta_1 = n_2 \sin \theta_2\).

Given \(n_1 = 1.00\) (air), \(\theta_1 = 30.0^\circ\).

Look up \(n_2 = 2.419\) for diamond (from Table 1.1).

Solve for \(\theta_2\).

Solution:

\[ \sin \theta_2 = \frac{n_1 \sin \theta_1}{n_2} = \frac{(1.00) \sin 30.0^\circ}{2.419} = \frac{0.500}{2.419} \approx 0.207 \]

\[ \theta_2 = \sin^{-1}(0.207) \approx 11.9^\circ \]

Significance:

Diamond causes a much larger change in direction (\(11.9^\circ\) vs. \(22.0^\circ\) for water) due to its significantly higher index of refraction.

This example reinforces that a greater difference in refractive indices leads to a more pronounced bending of light. This large bending effect in diamond is one reason for its famous sparkle.

1.4 Total Internal Reflection

• Occurs when light travels from a medium with a higher index of refraction to one with a lower index of refraction (\(n_1 > n_2\)).
• If the incident angle (\(\theta_1\)) exceeds a certain critical angle (\(\theta_c\)), all light is reflected back into the first medium.
• No light is refracted into the second medium.
• The reflected light still obeys the law of reflection (\(\theta_r = \theta_1\)).

Total internal reflection is a fascinating phenomenon where a “perfect mirror” effect is achieved without any metallic coating. It only happens under specific conditions, primarily when light tries to move from a denser medium to a less dense one at a sufficiently steep angle.

Critical Angle (\(\theta_c\))

• The incident angle at which the angle of refraction is \(90^\circ\).

• Using Snell’s Law (\(n_1 \sin \theta_1 = n_2 \sin \theta_2\)), with \(\theta_1 = \theta_c\) and \(\theta_2 = 90^\circ\):

  \[ n_1 \sin \theta_c = n_2 \sin 90^\circ \]

  \[ n_1 \sin \theta_c = n_2 \]

• Therefore, the critical angle is:

  \[ \theta_c = \sin^{-1} \left( \frac{n_2}{n_1} \right) \quad \text{for } n_1 > n_2 \]

The critical angle is a threshold. Below this angle, both reflection and refraction occur. At the critical angle, the refracted ray skims along the surface. Beyond the critical angle, only total internal reflection occurs.

Total Internal Reflection Diagram

Figure 1.14: (a) Refraction and reflection. (b) Critical angle, \(\theta_c\). (c) Total internal reflection.

Observe how the refracted ray moves further from the normal as the incident angle increases (a). At the critical angle (b), the refracted ray is parallel to the surface. Beyond that (c), all light is reflected internally.

Example: Critical Angle for Polystyrene

Question: What is the critical angle for light traveling in a polystyrene pipe (\(n_1 = 1.49\)) surrounded by air (\(n_2 = 1.00\))?

Strategy:

Confirm \(n_1 > n_2\).

Use the critical angle formula: \(\theta_c = \sin^{-1} \left( \frac{n_2}{n_1} \right)\).

Solution:

\[ \theta_c = \sin^{-1} \left( \frac{1.00}{1.49} \right) = \sin^{-1}(0.671) \approx 42.2^\circ \]

Significance:

Any light ray inside the plastic hitting the surface at an angle greater than \(42.2^\circ\) will be totally reflected, effectively turning the inside surface into a perfect mirror.

This calculation demonstrates why clear plastic can act as a mirror under certain conditions, a principle central to fiber optics. The critical angle is specific to the pair of materials involved.

Fiber Optics

• A key application of total internal reflection.
• Light is transmitted down thin fibers of plastic or glass.
• Light strikes the inner surface at an angle greater than \(\theta_c\), leading to continuous internal reflection.
• Fibers have a core (higher n) and cladding (lower n) to ensure reflection.

Figure 1.15: Light guided down an optical fiber by total internal reflection.

Fiber optics has revolutionized communication and medicine. The ability to guide light efficiently over long distances or around corners is due entirely to total internal reflection. The cladding prevents light from leaking between fibers.

Applications of Fiber Optics

Medical Endoscopes

• Explore body interiors without major surgery.
• Transmit light in, return images out.
• Used for diagnostics and microsurgery.

Figure 1.16: (a) Image transmission through optical fibers. (b) Endoscope image.

Telecommunications

• Transmit telephone, internet, and cable TV signals.
• Advantages:
  • Low loss: Light travels far without amplification.
  • High bandwidth: More data per fiber.
  • Reduced crosstalk: Signals don’t interfere.

The impact of fiber optics is immense. In medicine, it allows for minimally invasive procedures. In communications, it forms the backbone of the internet, enabling rapid and efficient data transfer across continents.

Diamonds and Total Internal Reflection

• Diamonds sparkle due to total internal reflection and high index of refraction (\(n_{diamond} = 2.419\)).
• Critical angle for diamond to air is very small: \(\theta_c = \sin^{-1}(1.00/2.419) \approx 24.4^\circ\).
• Light entering a diamond has trouble getting back out.
• Facets are cut to ensure multiple internal reflections, concentrating light before it exits.

Figure 1.19: Light trapped inside a diamond by total internal reflection, creating sparkle.

The small critical angle of diamond means that most light that enters it will undergo multiple internal reflections before exiting, which creates the brilliant sparkle we associate with diamonds. This is why diamond cutting is such a precise art, aimed at maximizing these internal reflections.

1.5 Dispersion

• Dispersion is the spreading of white light into its full spectrum of wavelengths (colors).
• Occurs because the index of refraction (n) for a given medium depends slightly on wavelength.
• Shorter wavelengths (e.g., violet) have a higher n and bend more than longer wavelengths (e.g., red).

Figure 1.20: Rainbow (a) and light dispersed by a prism (b) show the same spectrum of colors.

Dispersion is the reason we see rainbows and why prisms split white light into its constituent colors. The subtle variation of refractive index with wavelength, known as anomalous dispersion, is critical to this phenomenon.

Wavelength and Color

Figure 1.21: The visible light spectrum, showing colors associated with different wavelengths.

This spectrum illustrates the continuous nature of colors in visible light, ordered by wavelength. Red has the longest wavelength, and violet has the shortest.

Dispersion by a Prism

Figure 1.22: (a) A single wavelength refracted by a prism. (b) White light dispersed by a prism.

As white light enters the prism, each wavelength is refracted at a slightly different angle due to its unique index of refraction in the glass. This separates the colors, with violet bending most and red bending least, producing the spectrum.

Example: Dispersion by Crown Glass

Question: A beam of white light enters crown glass from air at an incidence angle of \(43.2^\circ\). What is the angle between the red (660 nm) and violet (410 nm) parts of the refracted light?

Strategy:

1. Find \(n\) for red and violet light in crown glass from Table 1.2.
   • \(n_{red} = 1.512\)
   • \(n_{violet} = 1.530\)
2. Use Snell’s Law (\(n_{air} \sin \theta_{air} = n \sin \theta\)) to find \(\theta_{red}\) and \(\theta_{violet}\).
   • \(n_{air} = 1.000\), \(\theta_{air} = 43.2^\circ\).
3. Calculate the difference: \(\Delta \theta = \theta_{red} - \theta_{violet}\).

Solution:

For red light:

\[ \theta_{red} = \sin^{-1} \left( \frac{1.000 \sin 43.2^\circ}{1.512} \right) \approx 27.0^\circ \]

For violet light:

\[ \theta_{violet} = \sin^{-1} \left( \frac{1.000 \sin 43.2^\circ}{1.530} \right) \approx 26.4^\circ \]

Difference:

\[ \Delta \theta = 27.0^\circ - 26.4^\circ = 0.6^\circ \]

Significance:

Even a small angular separation can become noticeable over long distances.

This calculation demonstrates how the small difference in the index of refraction for different wavelengths leads to measurable dispersion. This effect, though seemingly small here, is magnified in phenomena like rainbows.

Rainbows

• Formed by a combination of refraction and total internal reflection in water droplets.
• Light enters a drop, refracts, reflects from the back, and refracts again upon exiting.
• Since water disperses light, each color exits at a slightly different angle.
• Observer must look away from the Sun at a specific angle.

Figure 1.23: Light path in a water drop forming a rainbow.

Rainbows are a beautiful natural display of dispersion. The specific geometry of light entering, reflecting within, and exiting the spherical water droplets is what creates the distinct arc of colors.

Primary and Secondary Rainbows

Figure 1.24: (a) Different colors seen at different angles. (b) The arc of a rainbow. (c) A double rainbow.

1. Shows that different colors reach the observer’s eye from different droplets. (b) Explains why rainbows are arcs – the specific angle relative to the sun. (c) A secondary rainbow is rarer and results from two internal reflections, with colors reversed compared to the primary.

1.6 Huygens’s Principle

• A method to determine how and where waves propagate.
• States that every point on a wave front acts as a source of tiny, spherical wavelets.
• These wavelets spread out in the forward direction at the same speed as the wave.
• The new wave front is the surface tangent to all of these wavelets.

Figure 1.26: Huygens’s principle applied to a straight wave front.

Christiaan Huygens’s principle is a powerful conceptual tool for understanding wave propagation, especially when dealing with reflection, refraction, and diffraction. It allows us to visualize how waves move forward from one point in time to the next.

Explaining Reflection with Huygens’s Principle

• When a wave front strikes a mirror, each point on the wave front emits wavelets.
• Wavelets from points that hit the mirror first travel further.
• The tangent to these wavelets forms the new reflected wave front.
• This geometrically demonstrates that \(\theta_r = \theta_i\).

Figure 1.27: Huygens’s principle explains the law of reflection.

Huygens’s principle provides a beautiful and intuitive way to understand the law of reflection from a wave perspective, showing how the macroscopic behavior emerges from the microscopic propagation of wavelets.

Explaining Refraction with Huygens’s Principle

• When a wave front passes into a new medium, the speed of wavelets changes.
• If speed decreases (higher n), wavelets travel less distance in the same time.
• This causes the new wave front to change direction, bending towards the normal.
• This derivation leads directly to Snell’s Law.

Figure 1.28: Huygens’s principle explains the law of refraction.

Similarly, for refraction, Huygens’s principle reveals why light bends when changing media. The change in the speed of the wavelets in the new medium alters the direction of the overall wave front, providing a wave-based justification for Snell’s Law.

Diffraction

• Diffraction is the bending of waves around the edges of an opening or an obstacle.
• It is a fundamental wave characteristic.
• The amount of diffraction is most noticeable when the wavelength is comparable to the size of the opening/obstacle.

Figure 1.30: (a) Light acts like a ray. (b) Sound waves (larger wavelength) diffract.

Figure 1.31: Huygens’s principle illustrates diffraction through an opening.

The contrasting behavior of light and sound passing through a doorway highlights the role of wavelength in diffraction. Since light’s wavelength is tiny, its diffraction is usually negligible unless it encounters very small openings, which is when its wave nature becomes evident.

1.7 Polarization

• Polarization is the attribute that a wave’s oscillations have a definite direction relative to its propagation.
• For electromagnetic (EM) waves, polarization refers to the direction of the electric field oscillations.
• Unpolarized light has electric fields oscillating in random directions.
• Polarized light has electric fields oscillating in a specific direction (e.g., vertical, horizontal).

Figure 1.33: Direction of polarization is the direction of the electric field.

Polarization is a unique property of transverse waves like light. Unlike longitudinal waves, where oscillations are parallel to propagation, transverse waves can oscillate in different planes perpendicular to propagation. This difference is what polarization describes.

Polarizing Filters and Malus’s Law

• Polaroid materials act as polarizing slits, allowing only light with a specific polarization direction to pass.
• The polarization axis is the direction parallel to which the electric field passes.
• Malus’s Law: Describes the intensity of polarized light after passing through a polarizing filter. \[ I = I_0 \cos^2 \theta \]
  • \(I_0\): incident polarized light intensity.
  • \(I\): transmitted light intensity.
  • \(\theta\): angle between the direction of polarization and the filter’s axis.

Polarizing filters are made of long molecules aligned in one direction. They absorb electric field components parallel to the molecules and transmit those perpendicular to them, hence the polarization axis is perpendicular to the molecule alignment. Malus’s law is a quantitative way to predict the intensity reduction.

Malus’s Law Diagram

Figure 1.37: A polarizing filter transmits only the component of the wave parallel to its axis.

This diagram visually represents how a polarizing filter works. Only the component of the incident electric field that is aligned with the filter’s axis is transmitted, leading to a reduction in intensity proportional to \(\cos^2\theta\).

Example: Intensity Reduction by a Polarizing Filter

Question: What angle is needed between polarized light and a filter’s axis to reduce intensity by 90.0%?

Strategy:

Given \(I = 0.100 I_0\).

Use Malus’s Law: \(I = I_0 \cos^2 \theta\).

Solve for \(\theta\).

Solution:

\[ 0.100 I_0 = I_0 \cos^2 \theta \]

\[ \cos^2 \theta = 0.100 \]

\[ \cos \theta = \sqrt{0.100} \approx 0.3162 \]

\[ \theta = \cos^{-1}(0.3162) \approx 71.6^\circ \]

Significance: A large angle (\(71.6^\circ\)) is required for such a significant reduction, but at \(45^\circ\), intensity is already halved.

This example shows that Malus’s Law is not linear; a small change in angle near 0 degrees has less effect than a small change near 90 degrees. This non-linear response is characteristic of cosine squared relationships.

Polarization by Reflection

• When unpolarized light reflects off a surface, it becomes partially polarized.
• The reflected light is preferentially horizontally polarized.
• Polarizing sunglasses, with vertical axes, block much of this glare.
• Brewster’s Law: Reflected light is completely polarized at Brewster’s angle (\(\theta_b\)). \[ \tan \theta_b = \frac{n_2}{n_1} \] where \(n_1\) is the incident medium and \(n_2\) is the reflecting medium.
• At \(\theta_b\), the reflected and refracted rays are perpendicular.

Polarization by reflection is a common natural phenomenon. The reason reflected light is largely horizontally polarized is complex but can be intuitively understood as the vertical component of the electric field “sticking” to the surface more readily than the horizontal component, which “bounces off” more efficiently.

Polarization by Reflection Diagram

Figure 1.38: Unpolarized light reflects from a surface, becoming partially horizontally polarized.

The diagram clearly shows how the horizontal component of polarization is more strongly reflected, making the reflected light predominantly horizontally polarized. Brewster’s angle is a special case where this horizontal polarization is complete.

Polarization by Scattering

• Light scattered by air molecules (or other particles like smoke/dust) becomes partially polarized.
• Electrons in air molecules vibrate perpendicular to the incident light’s direction.
• They re-radiate light with polarization perpendicular to the original ray direction.
• This explains why the blue sky appears brighter or dimmer when viewed through polarizing sunglasses.

Figure 1.41: Polarization by scattering from air molecules.

Scattering is another natural process that polarizes light. The blue color of the sky is due to scattering, and the polarization observed in scattered light confirms the transverse wave nature of light. Photographers use polarizing filters to enhance sky contrast.

Liquid Crystals and Optical Activity

• Liquid Crystal Displays (LCDs): Based on the ability of liquid crystals to rotate light’s polarization by 90°.
  • This rotation can be turned off by applying a voltage.
  • Used in screens (watches, phones, TVs) to create contrast.
• Optical Activity: Some substances (e.g., sugar water, insulin) rotate the plane of polarization of light passing through them.
  • Due to asymmetrical molecular shapes.
  • Used to measure concentrations or analyze molecular structures.

Figure 1.42: (a) LCD with no voltage (light passes). (b) LCD with voltage (light blocked).

LCD technology is a prime example of applying polarization effects. The ability of liquid crystals to either rotate or not rotate polarized light, depending on an applied voltage, is ingeniously used to create the pixels on our screens. Optical activity, meanwhile, is a valuable tool in chemistry and biology for substance analysis.

Key Takeaways

• Speed of Light: Constant c in vacuum, slower in matter, defined by index of refraction n = c/v.
• Ray Model: Light travels in straight lines (rays) when interacting with objects much larger than its wavelength.
• Reflection: Angle of incidence equals angle of reflection (\(\theta_i = \theta_r\)).
• Refraction: Bending of light due to change in speed; governed by Snell’s Law (\(n_1 \sin \theta_1 = n_2 \sin \theta_2\)).
• Total Internal Reflection: Occurs when light goes from higher to lower n and incident angle exceeds critical angle (\(\theta_c = \sin^{-1}(n_2/n_1)\)). Crucial for fiber optics and diamond sparkle.
• Dispersion: Spreading of white light into colors due to n depending on wavelength (e.g., rainbows, prisms).
• Huygens’s Principle: Every point on a wavefront is a source of wavelets; explains reflection, refraction, and diffraction.
• Polarization: Direction of electric field oscillations in an EM wave. Unpolarized light has random oscillations.
• Malus’s Law: Intensity of polarized light through a filter: \(I = I_0 \cos^2 \theta\).
• Polarization by Reflection/Scattering: Reflected light is preferentially horizontally polarized; scattered light is partially polarized. Brewster’s angle (\(\tan \theta_b = n_2/n_1\)) for complete polarization by reflection.

Key Equations

Equation	Description
\(c \approx 3.00 \times 10^8 \text{ m/s}\)	Speed of light in a vacuum
\(n = \frac{c}{v}\)	Index of refraction
\(\theta_r = \theta_i\)	Law of reflection
\(n_1 \sin \theta_1 = n_2 \sin \theta_2\)	Snell’s Law (Law of refraction)
\(\theta_c = \sin^{-1} \left( \frac{n_2}{n_1} \right)\)	Critical angle for total internal reflection (\(n_1 > n_2\))
\(I = I_0 \cos^2 \theta\)	Malus’s Law (intensity of polarized light through a filter)
\(\tan \theta_b = \frac{n_2}{n_1}\)	Brewster’s Law (angle for complete polarization by reflection)

Key Terms

Term	Definition
Speed of Light (c)	The speed at which light and all other electromagnetic radiation propagates in a vacuum, approximately \(3.00 \times 10^8 \text{ m/s}\).
Index of Refraction (n)	A dimensionless quantity that describes how fast light travels through a medium, defined as the ratio of the speed of light in vacuum to its speed in the medium (\(n = c/v\)).
Ray Model of Light	A model that describes light propagation as straight lines (rays), valid when light interacts with objects much larger than its wavelength.
Law of Reflection	States that the angle of reflection equals the angle of incidence (\(\theta_r = \theta_i\)), both measured relative to the normal to the surface.
Refraction	The bending of light as it passes from one transparent medium to another, caused by a change in the speed of light.
Snell’s Law	The mathematical relationship governing refraction: \(n_1 \sin \theta_1 = n_2 \sin \theta_2\).
Total Internal Reflection	The phenomenon where all incident light is reflected back into the first medium when it strikes a boundary with a second medium of lower index of refraction at an angle greater than the critical angle.
Critical Angle (\(\theta_c\))	The incident angle that produces an angle of refraction of \(90^\circ\).
Dispersion	The spreading of white light into its full spectrum of wavelengths, due to the dependence of the index of refraction on wavelength.
Huygens’s Principle	A method stating that every point on a wave front is a source of wavelets that spread out in the forward direction, with the new wave front being tangent to these wavelets.
Diffraction	The bending of a wave around the edges of an opening or an obstacle.
Polarization	The attribute that a wave’s oscillations have a definite direction relative to the direction of propagation of the wave (for EM waves, the direction of the electric field).
Unpolarized Light	Light consisting of waves with electric fields oscillating in random directions.
Polarizing Filter	A material that transmits only light with electric fields oscillating parallel to its polarization axis.
Malus’s Law	\(I = I_0 \cos^2 \theta\), describing the intensity of polarized light after passing through a polarizing filter.
Brewster’s Angle (\(\theta_b\))	The angle of incidence at which light reflected from a surface is completely polarized.