Optics - Dispersion Scattering and Polarisation

Dispersion and Scattering Introduction to Dispersion and Scattering Light doesn't always travel in a straight line or maintain its pristine properties when interacting with matter. Scattering refers to the deflection or redirection of electromagnetic waves when they interact with particles or obstacles, while dispersion describes how the properties of light (particularly its speed) depend on frequency or wavelength. Understanding these phenomena is essential for optics, fiber communications, and spectroscopy. Types of Scattering Different scattering processes occur depending on the nature of the interaction and the relative size of scattering particles compared to the light's wavelength. Thomson Scattering Thomson scattering occurs when electromagnetic waves interact with free electrons or charged particles. The electron oscillates in response to the electric field and re-radiates the energy. A key characteristic is that Thomson scattering is frequency-independent—the scattering intensity is the same regardless of the light's frequency. This makes Thomson scattering fundamentally a classical phenomenon based on the acceleration of charges. Compton Scattering Compton scattering is a quantum mechanical process where photons collide with electrons. Unlike Thomson scattering, the scattered photon has a different frequency than the incident photon. The energy difference goes into kinetic energy of the electron. This is frequency-dependent and requires a quantum treatment of light as particles (photons). Rayleigh Scattering Rayleigh scattering occurs when light scatters from particles much smaller than its wavelength. The scattered intensity is inversely proportional to the fourth power of the wavelength: $I \propto 1/\lambda^4$. This is why the sky appears blue (blue light scatters more than red) and why sunsets are red (red light passes through more of the atmosphere). Rayleigh scattering is elastic, meaning the light's frequency remains unchanged. Mie Scattering Mie scattering occurs when particles are comparable to or larger than the light's wavelength. This is more complex than Rayleigh scattering and the scattering intensity is less dependent on wavelength. Mie scattering explains why clouds and fog appear white—they scatter all wavelengths relatively equally. Like Rayleigh scattering, Mie scattering is elastic. Raman Scattering Raman scattering is inelastic scattering where the frequency of scattered light differs from the incident light because the light excites molecular vibrational or rotational modes. Some photons lose energy (Stokes scattering) while others gain energy (anti-Stokes scattering). This is particularly useful for spectroscopy and identifying molecular structures. Types of Dispersion Now we shift from scattering to dispersion—how light's speed and behavior depend on wavelength or frequency. Normal vs. Anomalous Dispersion Normal dispersion occurs in most transparent materials: the refractive index decreases with increasing wavelength. This means blue light (shorter wavelength) travels slower than red light (longer wavelength). Normal dispersion is the typical behavior in glass and most dielectrics away from absorption bands. Anomalous dispersion occurs near strong absorption bands of a material: the refractive index increases with increasing wavelength, which is the opposite of normal dispersion. This counterintuitive behavior occurs only in narrow spectral regions where the material strongly absorbs light. Material Dispersion vs. Waveguide Dispersion These represent two distinct sources of dispersion: Material dispersion arises from the wavelength dependence of the material's intrinsic refractive index $n(\lambda)$. As light of different wavelengths travels through a material, each travels at a slightly different speed. This is fundamental to the material itself. Waveguide dispersion arises from the geometry of an optical waveguide (such as an optical fiber). Even if the core material had no material dispersion, the waveguide structure causes different wavelengths to propagate at different velocities because of how the wave is confined. This depends on the propagation constant and the waveguide geometry, not just the material. Characterization of Dispersion The Abbe Number The Abbe number (or Abbe value) provides a simple measure of how much a material disperses light. It's defined using refractive indices at three standard wavelengths: $$V = \frac{nD - 1}{nF - nC}$$ where $nD$, $nF$, and $nC$ are the refractive indices at three standard wavelengths (the D, F, and C Fraunhofer lines). A high Abbe number means the material is less dispersive (refractive index varies less with wavelength), while a low Abbe number indicates strong dispersion. This is commonly used to classify optical glasses and design achromatic lenses. Group Velocity and Group Velocity Dispersion In a dispersive medium, different frequency components of light travel at different speeds. The group velocity represents the speed at which the envelope of a light pulse travels: $$vg = \frac{c}{n + \omega \frac{dn}{d\omega}}$$ where $n$ is the refractive index, $\omega$ is the angular frequency, and $c$ is the speed of light in vacuum. When $\frac{dn}{d\omega} < 0$ (normal dispersion), the group velocity is less than $c/n$. When $\frac{dn}{d\omega} > 0$ (anomalous dispersion), the group velocity can exceed $c/n$ or even be larger than $c$ in the frame of reference—though this doesn't violate relativity because information travels at the phase velocity, not the group velocity. The Dispersion Parameter The dispersion parameter $D$ quantifies how much a pulse spreads over a given distance. The sign convention is: $D < 0$ indicates normal (positive) dispersion $D > 0$ indicates anomalous (negative) dispersion Note: This sign convention can be confusing—a negative dispersion parameter corresponds to normal dispersion, and vice versa. Effects of Group Velocity Dispersion on Pulses Understanding how dispersion affects pulses is crucial for fiber-optic communications. Pulse Spreading in Normal Dispersion In a normally dispersive medium, higher-frequency components travel slower than lower-frequency components. When a pulse with a range of frequencies enters such a medium: Lower-frequency (red) components travel faster and move to the front of the pulse Higher-frequency (blue) components travel slower and move to the back of the pulse This creates a positively chirped or up-chirped pulse, where frequency increases toward the pulse's trailing edge The pulse envelope spreads out temporally—it becomes wider in time. Pulse Spreading in Anomalous Dispersion In an anomalously dispersive medium, the behavior reverses: higher-frequency components travel faster than lower-frequency components. This produces a negatively chirped or down-chirped pulse, where frequency decreases toward the pulse's trailing edge. Again, the pulse envelope spreads temporally. Impact on Communications In optical-fiber communication systems, temporal spreading caused by group velocity dispersion is a critical limitation. If pulses spread too much, adjacent pulses begin to overlap, causing intersymbol interference and errors in data transmission. This sets limits on either: The distance over which data can be transmitted reliably The data rate (bit rate) that can be used Or both, unless dispersion is compensated Managing dispersion is thus a central concern in fiber-optic system design. Polarisation Introduction to Polarisation While dispersion describes how light's speed depends on frequency, polarisation describes the direction and behavior of the electric field vector as light propagates. Light can be polarised in different ways, and controlling polarisation is essential in optics, displays, microscopy, and many other applications. Polarisation States Polarisation describes how the electric field vector $\vec{E}$ behaves perpendicular to the direction of propagation. Linear Polarisation In linear polarisation, the electric field vector oscillates in a single fixed direction in space. The magnitude of the field may vary sinusoidally with time, but its direction never changes. If light is polarised horizontally, the field oscillates only in the horizontal direction. Linear polarisation is the simplest and most commonly encountered polarisation state. Circular Polarisation In circular polarisation, the electric field vector rotates at a constant rate, tracing out a circle as time progresses. For a given position in space, the tip of the electric field vector traces a circle in the plane perpendicular to the propagation direction. Circular polarisation has a chirality—it can be either: Right-handed circular polarisation: The field vector rotates clockwise when viewing the light approaching you Left-handed circular polarisation: The field vector rotates counterclockwise when viewing the light approaching you The handedness convention can vary in different fields, so context matters. Elliptical Polarisation In elliptical polarisation, the electric field vector traces an ellipse (not a circle) in the plane perpendicular to propagation. This is the most general polarisation state and occurs when two orthogonal linear components have unequal amplitudes or an arbitrary phase difference between them. Both linear and circular polarisation are special cases of elliptical polarisation. Changing Polarisation: Birefringence and Optical Elements Birefringence Some materials are birefringent, meaning they have different refractive indices for light polarised in different directions. Specifically, they have two principal axes along which the refractive indices differ. Light polarised along each axis travels at a different speed. When linearly polarised light enters a birefringent material at an arbitrary angle (not aligned with a principal axis), it can be decomposed into components along the two principal axes. These components travel at different speeds and accumulate different phases. By the time they exit the material, they have a phase retardation between them. This phase difference can convert the polarisation state. Wave Plates (Retarders) Wave plates or retarders are optical elements that exploit birefringence to convert one polarisation state into another: A quarter-wave plate introduces a phase retardation of $\lambda/4$ (or 90°) between orthogonal components. If linear polarisation enters at 45° to the plate's axes, it emerges as circular polarisation. A half-wave plate introduces a phase retardation of $\lambda/2$ (or 180°). It can rotate the plane of linear polarisation. Wave plates are essential components for controlling and manipulating polarisation in optical systems. The Faraday Effect <extrainfo> The Faraday effect is the rotation of the plane of linear polarisation when light propagates through a material in the presence of a magnetic field. This rotation angle is proportional to the magnetic field strength and the distance traveled. The Faraday effect is fundamentally different from birefringence—it's called circular birefringence because it represents different refractive indices for right- and left-handed circularly polarised light. Importantly, the Faraday effect is non-reciprocal: if light travels back through the same path, the rotation doesn't reverse, which is useful for optical isolators. </extrainfo> Dichroic Media and Polarising Filters Dichroic media attenuate (absorb) one polarisation mode more strongly than the other. A common example is a polarising filter or polariser, which transmits light of one linear polarisation direction while absorbing the orthogonal component. Malus's Law quantifies transmission through an ideal linear polariser: $$I = I0 \cos^2 \theta$$ where $I0$ is the incident intensity, $I$ is the transmitted intensity, and $\theta$ is the angle between the incident light's polarisation direction and the polariser's transmission axis. At $\theta = 0°$, all light is transmitted. At $\theta = 90°$, no light is transmitted. At $\theta = 45°$, half the intensity is transmitted. Natural Light and Partial Polarisation Not all light is perfectly polarised. Real light from the sun or from thermal sources is generally unpolarised or partially polarised. Unpolarised Light Unpolarised light (also called natural light) consists of a completely random mixture of all possible linear polarisation directions in equal proportions. Over time, the electric field vector points in every direction in the plane perpendicular to propagation with equal probability. No polarisation filter can isolate a preferred direction from unpolarised light—all polarisers transmit the same fraction regardless of their orientation. Partially Polarised Light Partially polarised light can be described as a superposition of two components: A completely unpolarised component A completely polarised component The degree of polarisation ranges from 0 (fully unpolarised) to 1 (fully polarised). Real light from the sun and from most ordinary sources is partially polarised due to the incoherent mixture of light from many independent emission events. Polarisation from Reflection and Scattering Reflected light often becomes partially polarised. When light is reflected from a smooth surface (like water) at a non-normal incidence angle, the component polarised parallel to the plane of incidence (p-polarised) and the component perpendicular to the plane of incidence (s-polarised) are reflected with different intensities. The reflected light is therefore partially polarised. Similarly, scattering of sunlight by atmospheric molecules (such as nitrogen and oxygen) preferentially scatters certain polarisation directions, making skylight partially polarised. This is why a polarising filter can reduce glare in photography—it blocks the partially polarised component from the sky while allowing the (relatively unpolarised) light from the object itself. Polarisation at Interfaces: Fresnel Equations and Brewster's Angle When light is incident on an interface between two media (such as air and glass), part of it reflects and part of it refracts. The Fresnel equations determine the reflected and transmitted amplitudes. The Fresnel Equations The Fresnel equations give the amplitude reflection and transmission coefficients for light incident on a planar interface. They differ for two orthogonal polarisation modes: s-polarised light (or TE): Electric field perpendicular to the plane of incidence p-polarised light (or TM): Electric field parallel to the plane of incidence The plane of incidence is defined by the incident ray and the normal to the surface. The Fresnel equations show that the reflected and transmitted fractions depend on: The angle of incidence The refractive indices of both media The polarisation of the light Remarkably, these two polarisation modes behave differently at interfaces, which is the source of phenomena like Brewster's angle. Brewster's Angle At a special angle called Brewster's angle, the reflected light for p-polarised light becomes zero. All of the p-polarised component is transmitted (refracted) into the second medium—none is reflected. At Brewster's angle $\thetaB$: $$\tan \thetaB = \frac{n2}{n1}$$ where $n1$ is the refractive index of the incident medium and $n2$ is the refractive index of the transmitted medium. At Brewster's angle, the reflected light is perfectly polarised in the s-polarisation direction (perpendicular to the plane of incidence), because p-polarised light has zero reflection. This is why polarising sunglasses with the transmission axis aligned with s-polarisation can effectively reduce glare from horizontal surfaces like roads or water—Brewster's angle for air-to-surface reflection is roughly 56° for typical surfaces, which is close to the typical angle at which light reflects from horizontal surfaces into the eyes. Summary Dispersion and scattering describe how light's speed and direction change in matter, with different mechanisms (Thomson, Compton, Rayleigh, Mie, and Raman scattering) governing different regimes. Dispersion's effects on pulses are critical for fiber communications. Polarisation describes the directional behavior of the electric field and can be controlled with birefringent elements, manipulated using Malus's law, and understood at interfaces through the Fresnel equations and Brewster's angle. Together, these concepts form the foundation for understanding light-matter interactions in optics.