10.4 Brewster's angle

Brewster's angle is a fascinating optical phenomenon where light with a specific polarization is perfectly transmitted through a transparent surface. It occurs when the reflected and refracted rays are perpendicular, resulting in no reflection for one polarization.

This concept is crucial in understanding how light interacts with different materials. It has practical applications in polarizing filters, glare reduction, and microscopy techniques. Brewster's angle depends on the refractive indices of the materials involved and is independent of wavelength.

Definition of Brewster's angle

• Brewster's angle is the angle of incidence at which light with a particular polarization is perfectly transmitted through a transparent dielectric surface, with no reflection
• When unpolarized light is incident at this angle, the light that is reflected from the surface is therefore perfectly polarized
• Brewster's angle is named after the Scottish physicist Sir David Brewster who first described this phenomenon in 1812

Relationship to polarization

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• Light can be polarized in different ways, such as linear, circular, or elliptical polarization
• At Brewster's angle, the reflected light is linearly polarized in the plane perpendicular to the plane of incidence
  • This plane is known as the s-polarization plane (from the German "senkrecht", meaning perpendicular)
• The refracted light is partially polarized in the plane parallel to the plane of incidence, known as p-polarization
• The degree of polarization depends on the refractive indices of the media involved

Snell's law and refractive indices

• Snell's law describes the relationship between the angles of incidence and refraction when light passes through a boundary between two different isotropic media
• The law states that the ratio of the sines of the angles of incidence and refraction is equivalent to the ratio of the refractive indices of the two media
  • Mathematically: $\frac{\sin \theta_1}{\sin \theta_2} = \frac{[n_2](https://www.fiveableKeyTerm:n_2)}{[n_1](https://www.fiveableKeyTerm:n_1)}$, where $\theta_1$ and $\theta_2$ are the angles of incidence and refraction, and $n_1$ and $n_2$ are the refractive indices of the respective media
• The refractive index of a medium is a dimensionless number that describes how fast light propagates through the material compared to vacuum
• Snell's law is used in the derivation and calculation of Brewster's angle

Derivation of Brewster's angle

• The derivation of Brewster's angle involves the use of Fresnel equations and the conditions for zero reflection
• Brewster's angle occurs when the reflected and refracted rays are perpendicular to each other

Fresnel equations

• Fresnel equations describe the behavior of light when it strikes a boundary between two different media
• They relate the amplitude and phase of the reflected and transmitted waves to the amplitude and phase of the incident wave
• There are separate equations for s-polarized and p-polarized light
  • For s-polarized light: $r_s = \frac{n_1 \cos \theta_1 - n_2 \cos \theta_2}{n_1 \cos \theta_1 + n_2 \cos \theta_2}$ and $t_s = \frac{2n_1 \cos \theta_1}{n_1 \cos \theta_1 + n_2 \cos \theta_2}$
  • For p-polarized light: $r_p = \frac{n_2 \cos \theta_1 - n_1 \cos \theta_2}{n_2 \cos \theta_1 + n_1 \cos \theta_2}$ and $t_p = \frac{2n_1 \cos \theta_1}{n_2 \cos \theta_1 + n_1 \cos \theta_2}$

Reflection and transmission coefficients

• The Fresnel equations give the reflection and transmission coefficients, which are the ratios of the amplitudes of the reflected and transmitted waves to the amplitude of the incident wave
• The reflection coefficients ($r_s$ and $r_p$) determine the amount of light reflected at the interface for each polarization
• The transmission coefficients ($t_s$ and $t_p$) determine the amount of light transmitted through the interface for each polarization
• The reflectance and transmittance, which are the fractions of the incident power that are reflected and transmitted, are obtained by squaring the absolute values of the respective coefficients

Conditions for zero reflection

• Brewster's angle is derived by setting the reflection coefficient for p-polarized light ($r_p$) equal to zero
• This condition leads to the equation: $\tan \theta_B = \frac{n_2}{n_1}$, where $\theta_B$ is Brewster's angle, and $n_1$ and $n_2$ are the refractive indices of the first and second media, respectively
• When the angle of incidence equals Brewster's angle, the reflected and refracted rays are perpendicular to each other, and the reflection of p-polarized light is minimized

Properties of Brewster's angle

• Brewster's angle has several unique properties that make it useful in various applications

Dependence on materials

• The value of Brewster's angle depends on the refractive indices of the materials forming the interface
• For an interface between two transparent dielectric materials with refractive indices $n_1$ and $n_2$, Brewster's angle is given by: $\theta_B = \arctan \left(\frac{n_2}{n_1}\right)$
• Examples:
  • For an air-water interface ($n_{air} \approx 1$, $n_{water} \approx 1.33$), Brewster's angle is approximately 53°
  • For an air-glass interface ($n_{air} \approx 1$, $n_{glass} \approx 1.5$), Brewster's angle is approximately 56°

Wavelength independence

• Brewster's angle is independent of the wavelength of the incident light, as long as the refractive indices of the materials are not strongly dependent on wavelength (i.e., the materials are not dispersive)
• This property allows Brewster's angle to be used for broadband applications, such as polarizing filters and Brewster windows

Polarizing angle vs critical angle

• Brewster's angle is sometimes called the polarizing angle because it produces polarized reflected light
• It should not be confused with the critical angle, which is the angle of incidence above which total internal reflection occurs
• The critical angle is given by: $\theta_c = \arcsin \left(\frac{n_2}{n_1}\right)$, where $n_1 > n_2$
• While Brewster's angle is related to polarization, the critical angle is related to total internal reflection and is used in applications such as optical fibers and prisms

Applications of Brewster's angle

• Brewster's angle has numerous applications in optics, photonics, and related fields

Polarizing filters and Brewster windows

• Polarizing filters based on Brewster's angle are used to produce linearly polarized light
• These filters consist of a stack of dielectric plates oriented at Brewster's angle relative to the incident light
  • Each interface reflects s-polarized light and transmits p-polarized light, resulting in a highly polarized transmitted beam
• Brewster windows are used in laser systems to minimize reflections and losses at the interface between the gain medium and the surrounding environment
  • The windows are oriented at Brewster's angle to the laser beam, ensuring maximum transmission of the desired polarization

Glare reduction techniques

• Brewster's angle is used in glare reduction techniques for various applications, such as sunglasses, camera lenses, and display screens
• By orienting the surface at Brewster's angle relative to the incident light, the reflection of p-polarized light is minimized, reducing glare and improving visibility
• Polarized sunglasses exploit this principle by blocking s-polarized light, which is the main component of glare from horizontal surfaces like water and roads

Brewster angle microscopy

• Brewster angle microscopy is a technique used to study thin films and surfaces by exploiting the properties of Brewster's angle
• In this technique, a p-polarized laser beam is incident on the sample at Brewster's angle, and the reflected light is detected
• Changes in the sample's refractive index or thickness result in changes in the reflected intensity, allowing for high-sensitivity measurements of surface properties and thin film growth

Experimental verification

• Experimental verification of Brewster's angle involves measuring the angle at which the reflection of p-polarized light is minimized and using this information to determine the refractive indices of the materials

Measuring Brewster's angle

• To measure Brewster's angle, an experimental setup typically consists of a laser source, a polarizer, the sample (e.g., a dielectric interface), and a detector
• The laser beam is linearly polarized using the polarizer and directed onto the sample at various angles of incidence
• The intensity of the reflected light is measured using the detector as a function of the angle of incidence
• Brewster's angle is identified as the angle at which the reflected intensity is minimized for p-polarized light

Determining refractive indices

• Once Brewster's angle is measured, the refractive indices of the materials can be determined using the equation: $\tan \theta_B = \frac{n_2}{n_1}$
• If one of the refractive indices is known (e.g., air), the other refractive index can be calculated from the measured Brewster's angle
• This technique is useful for characterizing the optical properties of materials, particularly thin films and substrates

Polarization state analysis

• Experimental verification of Brewster's angle also involves analyzing the polarization state of the reflected and transmitted light
• By measuring the intensity of the reflected and transmitted light for different polarization orientations (e.g., using a rotating analyzer), the degree of polarization and the polarization angle can be determined
• This information helps to confirm the theoretical predictions of the Fresnel equations and the properties of Brewster's angle

Limitations and special cases

• While Brewster's angle is a powerful concept in optics, there are some limitations and special cases to consider

Absorbing media and complex refractive indices

• The derivation of Brewster's angle assumes that the media are non-absorbing and have real refractive indices
• When one or both media are absorbing (e.g., metals or semiconductors), the refractive indices become complex, with a real part (n) and an imaginary part (k)
• In this case, the concept of Brewster's angle becomes more complicated, as the reflectance of p-polarized light does not necessarily go to zero at a single angle
• The pseudo-Brewster angle is defined as the angle of incidence at which the reflectance of p-polarized light is minimized, but not necessarily zero

Anisotropic materials

• Anisotropic materials, such as crystals and liquid crystals, have refractive indices that depend on the direction of light propagation and polarization
• In these materials, the concept of Brewster's angle becomes more complex, as there may be multiple Brewster angles depending on the orientation of the optic axis relative to the interface
• The calculation of Brewster's angle in anisotropic materials requires the use of more advanced formalisms, such as the 4x4 matrix method or the Berreman method

Non-planar interfaces and curved surfaces

• The derivation of Brewster's angle assumes a planar interface between two media
• When the interface is non-planar or curved (e.g., a lens surface), the local angle of incidence varies across the surface
• In these cases, the Brewster effect may not be uniform across the entire interface, leading to a more complex distribution of reflected and transmitted light
• Modeling the behavior of light at non-planar interfaces requires the use of more advanced techniques, such as ray tracing or finite-difference time-domain (FDTD) simulations