Module 5 - Summary

Summary Notes Module 5

• In Electromagnetic Optics, light is described as a vector.

• An electromagnetic field is described by 2 vector fields that are dependent on space, \( \mathbf{r} = (x, y, z) \), and time, \( t \).

• Electric field: \( \mathbf{E}(\mathbf{r}, t) = (E_x, E_y, E_z) \)
• Magnetic field: \( \mathbf{H}(\mathbf{r}, t) = (H_x, H_y, H_z) \)

• Maxwell’s equation in free-space (\( n = 1 \)):

• \( \nabla \times \mathbf{H} = \frac{\partial \mathbf{D}}{\partial t} = \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} \), where \( \epsilon_0 = \frac{1}{36 \pi} \times 10^{-9} \, \text{F/m} \) is the electric permittivity.
• \( \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \), where \( \mu_0 = 4 \pi \times 10^7 \, \text{H/m} \) is the magnetic permeability.
• \( \nabla \cdot \mathbf{E} = 0 \)
• \( \nabla \cdot \mathbf{H} = 0 \)

Each component of \( \mathbf{E}(\mathbf{r}, t) \) and \( \mathbf{H}(\mathbf{r}, t) \) should satisfy the wave equation (i.e., Module 2). For example, considering \( E_y \), then:

\( \nabla^2 E_y(\mathbf{r}, t) - \frac{1}{v^2} \frac{\partial^2 E_y(\mathbf{r}, t)}{\partial t^2} = 0 \)

where \( v = \frac{c}{n} \) and \( \nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} \) is the Laplacian operator in Cartesian coordinates.

• In a medium with no free electric and magnetic charges, there are 4 vectors:

• Electric field: \( \mathbf{E}(\mathbf{r}, t) = (E_x, E_y, E_z) \)
• Electric displacement field: \( \mathbf{D}(\mathbf{r}, t) = (D_x, D_y, D_z) \)
• Magnetic field: \( \mathbf{H}(\mathbf{r}, t) = (H_x, H_y, H_z) \)
• Magnetic flux density field: \( \mathbf{B}(\mathbf{r}, t) = (B_x, B_y, B_z) \)

where \( \mathbf{D} = \epsilon_0 \mathbf{E} + \mathbf{P} \) and \( \mathbf{B} = \mu_0 \mathbf{H} + \mathbf{M} \). Note that \( \mathbf{P} = \mathbf{M} = 0 \) in free space. Both \( \mathbf{P} \) (i.e., polarization density field) and \( \mathbf{M} \) (i.e., magnetization density field) are dependent on the medium.

• Maxwell’s equation in a medium (\( n > 1 \)):

• \( \nabla \times \mathbf{H} = \frac{\partial \mathbf{D}}{\partial t} \)
• \( \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \)
• \( \nabla \cdot \mathbf{D} = 0 \)
• \( \nabla \cdot \mathbf{B} = 0 \)

• The Poynting vector, \( \mathbf{S} = \mathbf{E} \times \mathbf{H} \), represents the flow of electromagnetic power. The Poynting vector is perpendicular to \( \mathbf{E} \) and \( \mathbf{H} \). For example, if \( \mathbf{E}(\mathbf{r}, t) = (E_x, 0, 0) \) and \( \mathbf{H}(\mathbf{r}, t) = (0, H_y, 0) \), then the Poynting vector only has a unique component, \( \mathbf{S} = \mathbf{E} \times \mathbf{H} = \begin{vmatrix} \hat{x} & \hat{y} & \hat{z} \\ E_x & 0 & 0 \\ 0 & H_y & 1 \end{vmatrix} = \hat{z} E_x H_y = (0, 0, E_x H_y) \), so \( S_z = E_x H_y \).

• Medium so \( \mathbf{E} \) and \( \mathbf{P} \) are related each other, and \( \mathbf{H} \) and \( \mathbf{M} \) are related each other.

Cases:

• \( \mathbf{E} \) and \( \mathbf{P} \) are linearly related → medium is dielectric
• \( \mathbf{E} \) and \( \mathbf{P} \) are invariant to space → homogeneous medium
• \( \mathbf{E} \) and \( \mathbf{P} \) are parallel → isotropic medium
• \( \mathbf{E}(t_1) \) determines \( \mathbf{P}(t_1) \) → nondispersive medium
• \( \mathbf{E}(\mathbf{r}_1) \) determines \( \mathbf{P}(\mathbf{r}_1) \) → spatially nondispersive medium

If a medium is linear, nondispersive, homogeneous, and isotropic, then \( \mathbf{P} = \epsilon_0 \chi \mathbf{E} \) and \( \mathbf{D} = \epsilon \mathbf{E} \), where \( \chi \) is the medium's electric susceptibility, and \( \epsilon = \epsilon_0(1 + \chi) \) is the medium’s electric permittivity. Similarly, \( \mathbf{M} = \mu_0 \chi_m \mathbf{H} \) and \( \mathbf{B} = \mu \mathbf{H} \), where \( \chi_m \) is the medium’s magnetic susceptibility, and \( \mu = \mu_0(1 + \chi_m) \) is the medium’s magnetic permeability.

• Simplification of Maxwell’s equations:

• \( \nabla \times \mathbf{H} = \frac{\partial \mathbf{D}}{\partial t} = \epsilon \frac{\partial \mathbf{E}}{\partial t} \)
• \( \nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t} = - \mu \frac{\partial \mathbf{H}}{\partial t} \)
• \( \nabla \cdot \mathbf{E} = 0 \)
• \( \nabla \cdot \mathbf{H} = 0 \)

where \( v = \frac{1}{\sqrt{\epsilon \mu}} \), \( c = \frac{1}{\sqrt{\epsilon_0 \mu_0}} \), and \( n = \sqrt{1 + \chi} \) (i.e., \( \mu = \mu_0 \), medium without magnetic properties).

If \( n = \sqrt{1 + \chi} \), then \( \chi = n^2 - 1 \), the electric susceptibility can be estimated from the medium’s refractive index. In general, if the medium has electric and magnetic properties, \( n = \sqrt{(1 + \chi)(1 + \chi_m)} \)

• Monochromatic electromagnetic waves:

• \( \mathbf{E}(\mathbf{r}, t) = \text{Re}[\mathbf{E}(\mathbf{r}) e^{j \omega t}] \)
• \( \mathbf{D}(\mathbf{r}, t) = \text{Re}[\mathbf{D}(\mathbf{r}) e^{j \omega t}] \)
• \( \mathbf{H}(\mathbf{r}, t) = \text{Re}[\mathbf{H}(\mathbf{r}) e^{j \omega t}] \)
• \( \mathbf{B}(\mathbf{r}, t) = \text{Re}[\mathbf{B}(\mathbf{r}) e^{j \omega t}] \)

The Maxwell's equations for monochromatic electromagnetic waves:

• \( \nabla \times \mathbf{H} = j \omega \mathbf{D} \)
• \( \nabla \times \mathbf{E} = - j \omega \mathbf{B} \)
• \( \nabla \cdot \mathbf{D} = 0 \)
• \( \nabla \cdot \mathbf{B} = 0 \)

• Absorption – decrease of light intensity through propagation:

• Beer’s law: \( I(x) = I_0 e^{-\alpha x} \), where \( I(x) \) is the transmitted intensity after the light travels a distance \( x \), \( I_0 \) is the incident light intensity, and \( \alpha \) is the linear attenuation coefficient (units: m\(^{-1}\)). We can define the penetration depth as the inverse of the attenuation coefficient: \( \delta = \frac{1}{\alpha} \), which is the distance in which the intensity of the transmitted light is reduced by a factor of \( 1/e \).
• Transmittance: \( T = \frac{I(x)}{I_0} = e^{-\alpha x} \) and Absorptance: \( A = \log_{10}(T) = 0.4343 \alpha x \)

• The complex refractive index when the susceptibility is complex (\( \chi = \chi' + j \chi'' \)): \( n - j \frac{\alpha}{2k_0} = \sqrt{1 + \chi' + j \chi''} \), where \( k_0 = \frac{2\pi}{\lambda_0} \) is the free-space wavenumber.

• Dispersion occurs when the refractive index changes with the light’s wavelength:

• Abbe number
• Cauchy’s equation
• Sellmeier’s equation