10.3 Fresnel equations

Fresnel equations are key to understanding how light behaves at the boundary between different materials. They describe how much light is reflected or transmitted when it hits a surface, depending on factors like the angle and polarization of the incoming light.

These equations are crucial for many optical applications, from designing anti-reflective coatings to creating polarizing filters. They help us predict and control how light interacts with various materials, making them essential tools in optics and photonics.

Fresnel equations overview

• Fresnel equations describe the behavior of electromagnetic waves at the interface between two different media
• They relate the amplitudes of the reflected and transmitted waves to the amplitude of the incident wave
• Understanding Fresnel equations is crucial for analyzing the propagation of light through various optical systems

Reflection and transmission coefficients

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• Reflection coefficient ($r$) represents the ratio of the reflected wave amplitude to the incident wave amplitude
• Transmission coefficient ($t$) represents the ratio of the transmitted wave amplitude to the incident wave amplitude
• Both coefficients depend on the polarization state of the incident wave and the angle of incidence
• The coefficients are complex quantities, indicating that the reflected and transmitted waves may experience a phase shift

Polarization states

• Fresnel equations consider two orthogonal polarization states: s-polarization (perpendicular to the plane of incidence) and p-polarization (parallel to the plane of incidence)
• The reflection and transmission coefficients differ for s-polarized and p-polarized waves
• The polarization state of the incident wave affects how much of the wave is reflected or transmitted at the interface
• Polarization-dependent effects, such as Brewster's angle and total internal reflection, can be explained using Fresnel equations

Derivation of Fresnel equations

• The derivation of Fresnel equations relies on applying Maxwell's equations and boundary conditions at the interface between two media
• It involves considering the continuity of the tangential components of the electric and magnetic fields across the interface
• The derivation also incorporates Snell's law, which relates the angles of incidence, reflection, and transmission

Maxwell's equations at interface

• Maxwell's equations (Gauss's law, Faraday's law, Ampère's law, and the absence of magnetic monopoles) form the foundation for deriving Fresnel equations
• The equations are applied to the electric and magnetic fields at the interface between two media
• The continuity of the tangential components of the electric and magnetic fields is a consequence of Maxwell's equations

Boundary conditions

• Boundary conditions ensure that the electric and magnetic fields behave consistently at the interface
• The tangential components of the electric field must be continuous across the interface
• The normal component of the electric displacement field must be continuous across the interface
• The tangential components of the magnetic field must be continuous across the interface

Snell's law

• Snell's law relates the angles of incidence, reflection, and transmission at the interface between two media
• It states that $n_1 \sin \theta_1 = n_2 \sin \theta_2$, where $n_1$ and $n_2$ are the refractive indices of the two media, and $\theta_1$ and $\theta_2$ are the angles of incidence and transmission, respectively
• Snell's law is used in the derivation of Fresnel equations to relate the wave vectors of the incident, reflected, and transmitted waves

Fresnel equations for dielectrics

• Dielectrics are non-conducting materials that can be polarized by an external electric field
• Fresnel equations for dielectrics describe the reflection and transmission of light at the interface between two dielectric media
• The equations take different forms depending on the angle of incidence and the polarization state of the incident wave

Normal incidence

• Normal incidence occurs when the incident wave is perpendicular to the interface ($\theta_1 = 0$)
• At normal incidence, the reflection and transmission coefficients for both s-polarized and p-polarized waves are given by:
  • $r = \frac{n_1 - n_2}{n_1 + n_2}$
  • $t = \frac{2n_1}{n_1 + n_2}$
• The reflectance ($R$) and transmittance ($T$) can be calculated as $R = |r|^2$ and $T = \frac{n_2}{n_1}|t|^2$

Oblique incidence

• Oblique incidence occurs when the incident wave makes a non-zero angle with the normal to the interface
• The Fresnel equations for oblique incidence are more complex and depend on the polarization state:
  • For s-polarization: $r_s = \frac{n_1 \cos \theta_1 - n_2 \cos \theta_2}{n_1 \cos \theta_1 + n_2 \cos \theta_2}$, $t_s = \frac{2n_1 \cos \theta_1}{n_1 \cos \theta_1 + n_2 \cos \theta_2}$
  • For p-polarization: $r_p = \frac{n_2 \cos \theta_1 - n_1 \cos \theta_2}{n_2 \cos \theta_1 + n_1 \cos \theta_2}$, $t_p = \frac{2n_1 \cos \theta_1}{n_2 \cos \theta_1 + n_1 \cos \theta_2}$
• The angles $\theta_1$ and $\theta_2$ are related by Snell's law

Brewster's angle

• Brewster's angle is a special angle of incidence at which the reflected p-polarized light vanishes
• It occurs when the reflected and transmitted rays are perpendicular to each other
• Brewster's angle ($\theta_B$) is given by $\tan \theta_B = \frac{n_2}{n_1}$
• At Brewster's angle, the reflected light is entirely s-polarized, while the transmitted light is a mixture of s-polarized and p-polarized components

Fresnel equations for conductors

• Conductors are materials that allow the flow of electric current
• Fresnel equations for conductors describe the reflection and transmission of light at the interface between a dielectric and a conducting medium
• The equations for conductors are more complex due to the presence of free charge carriers and the resulting absorption of light

Complex refractive index

• Conductors are characterized by a complex refractive index, $\tilde{n} = n + i\kappa$, where $n$ is the real part and $\kappa$ is the imaginary part (extinction coefficient)
• The real part determines the phase velocity of light in the medium, while the imaginary part is related to the absorption of light
• The complex refractive index is used in the Fresnel equations for conductors to account for the attenuation of the transmitted wave

Reflection coefficients

• The Fresnel equations for conductors provide the reflection coefficients for s-polarized and p-polarized waves:
  • For s-polarization: $r_s = \frac{\cos \theta_1 - \sqrt{\tilde{n}^2 - \sin^2 \theta_1}}{\cos \theta_1 + \sqrt{\tilde{n}^2 - \sin^2 \theta_1}}$
  • For p-polarization: $r_p = \frac{\tilde{n}^2 \cos \theta_1 - \sqrt{\tilde{n}^2 - \sin^2 \theta_1}}{\tilde{n}^2 \cos \theta_1 + \sqrt{\tilde{n}^2 - \sin^2 \theta_1}}$
• The transmission coefficients are not typically considered for conductors since the transmitted wave is strongly attenuated

Skin depth

• The skin depth ($\delta$) is the distance over which the amplitude of the transmitted wave decays by a factor of $1/e$ in a conductor
• It is given by $\delta = \frac{\lambda}{2\pi\kappa}$, where $\lambda$ is the wavelength of the incident light and $\kappa$ is the imaginary part of the complex refractive index
• The skin depth is typically very small for conductors (on the order of nanometers), indicating that light is mostly reflected and hardly penetrates the conductor

Reflectance and transmittance

• Reflectance ($R$) is the fraction of the incident light intensity that is reflected at the interface
• Transmittance ($T$) is the fraction of the incident light intensity that is transmitted through the interface
• Both reflectance and transmittance depend on the angle of incidence and the polarization state of the incident light

Reflectance vs angle of incidence

• The reflectance varies with the angle of incidence according to the Fresnel equations
• For dielectrics, the reflectance increases with increasing angle of incidence, reaching a maximum value at grazing incidence ($\theta_1 = 90^\circ$)
• At Brewster's angle, the reflectance for p-polarized light drops to zero, while the reflectance for s-polarized light remains non-zero
• For conductors, the reflectance is generally high and varies less with the angle of incidence compared to dielectrics

Transmittance vs angle of incidence

• The transmittance decreases with increasing angle of incidence, as more light is reflected at larger angles
• For dielectrics, the transmittance reaches a minimum value at grazing incidence
• At Brewster's angle, the transmittance for p-polarized light reaches a maximum, while the transmittance for s-polarized light is reduced
• For conductors, the transmittance is typically very low due to the strong absorption of light in the medium

Energy conservation

• The principle of energy conservation requires that the sum of reflectance and transmittance equals unity ($R + T = 1$) for non-absorbing media
• For absorbing media, such as conductors, the sum of reflectance, transmittance, and absorptance ($A$) equals unity ($R + T + A = 1$)
• The Fresnel equations satisfy energy conservation, ensuring that the total energy of the incident, reflected, and transmitted waves is conserved

Applications of Fresnel equations

• Fresnel equations have numerous applications in optics and photonics, enabling the design and analysis of various optical systems and devices
• They are used to calculate the reflection and transmission properties of materials, optimize optical coatings, and control the polarization state of light

Optical coatings

• Optical coatings are thin layers of materials deposited on the surface of optical components to modify their reflection and transmission properties
• Anti-reflection coatings reduce the reflectance by destructive interference of the reflected waves from multiple interfaces
• High-reflection coatings increase the reflectance by constructive interference of the reflected waves
• The design of optical coatings relies on the Fresnel equations to determine the optimal layer thicknesses and refractive indices

Thin film interference

• Thin film interference occurs when light reflects from the top and bottom surfaces of a thin film, leading to interference effects
• The Fresnel equations are used to calculate the reflection and transmission coefficients at each interface
• The interference pattern depends on the film thickness, refractive index, and angle of incidence
• Applications of thin film interference include anti-reflection coatings, dichroic filters, and Fabry-Pérot interferometers

Polarizing filters

• Polarizing filters are devices that selectively transmit light of a specific polarization state while blocking light of the orthogonal polarization
• The Fresnel equations predict the existence of Brewster's angle, which is exploited in the design of polarizing filters
• At Brewster's angle, the reflected light is entirely s-polarized, while the transmitted light is predominantly p-polarized
• Polarizing filters are used in various applications, such as glare reduction, 3D glasses, and liquid crystal displays (LCDs)

Limitations and extensions

• The Fresnel equations in their basic form have certain limitations and may need to be extended to account for more complex scenarios
• These limitations include the assumption of perfectly smooth and flat interfaces, isotropic media, and linear optics

Rough surfaces

• Real surfaces are not perfectly smooth and may have random or periodic roughness
• Rough surfaces can scatter light in various directions, reducing the specular reflection and transmission described by the Fresnel equations
• Modified Fresnel equations or numerical methods (Rayleigh-Rice theory, Beckmann-Kirchhoff theory) are used to model the reflection and transmission from rough surfaces
• The effect of surface roughness becomes more significant when the roughness scale is comparable to or larger than the wavelength of light

Anisotropic media

• Anisotropic media have direction-dependent optical properties, characterized by a tensor refractive index
• The Fresnel equations need to be generalized to account for the anisotropic nature of the media
• The reflection and transmission coefficients become more complex and depend on the orientation of the optic axis relative to the interface
• Examples of anisotropic media include birefringent crystals (calcite, quartz) and liquid crystals

Nonlinear optics

• Nonlinear optics deals with the interaction of light with matter at high intensities, where the optical response becomes nonlinear
• The Fresnel equations assume linear optics, where the polarization of the medium is proportional to the electric field
• In nonlinear optics, additional terms (second-order, third-order) contribute to the polarization, leading to phenomena such as second-harmonic generation, sum-frequency generation, and self-phase modulation
• Nonlinear Fresnel equations are used to describe the reflection and transmission of light in nonlinear media, taking into account the intensity-dependent refractive index and nonlinear susceptibility