Physics of Reflections and Refractions

Snell's Laws

• Illustrates the behavior of a wave (light) as it transitions between two media with different properties.

• Key parameters:

  • \theta_i: Angle of incidence.
  • \theta_r: Angle of reflection.
  • \theta_t: Angle of refraction.
  • k_i: Wave vector of the incident wave.
  • k_r: Wave vector of the reflected wave.
  • k_t: Wave vector of the transmitted wave.
  • \epsilon1, \mu1: Permittivity and permeability of medium 1.
  • \epsilon2, \mu2: Permittivity and permeability of medium 2.

Angles of Incidence, Reflection & Refraction

• Snell's Law of Reflection: The angle of reflection equals the angle of incidence.

  • \thetai = \thetar (8.28a)

• Snell's Law of Refraction: Relates the sine of the angles of incidence and refraction to the ratio of phase velocities in the two media.

  • \frac{\sin \thetat}{\sin \thetai} = \sqrt{\frac{\mu1 \epsilon1}{\mu2 \epsilon2}} = \frac{v{p1}}{v{p2}} (8.28b)

Nonmagnetic Media

• Index of Refraction (n): Defined as the ratio of the speed of light in vacuum (c) to the phase velocity (v_p) in the medium.

  • n = \frac{c}{vp} = \sqrt{\mur \epsilonr} = \sqrt{\frac{\mu \epsilon}{\mu0 \epsilon_0}} (8.29)

• Rewriting Snell's Law of Refraction: Using the index of refraction.

  • \frac{\sin \thetat}{\sin \thetai} = \frac{n1}{n2} = \sqrt{\frac{\mu{r1} \epsilon{r1}}{\mu{r2} \epsilon{r2}}} (8.30)

• For Nonmagnetic Materials (\mu{r1} = \mu{r2} = 1):

  • \frac{\sin \thetat}{\sin \thetai} = \frac{n1}{n2} = \sqrt{\frac{\epsilon{r1}}{\epsilon{r2}}} (8.31)

Refraction Behavior

• Inward Refraction:

  • Occurs when a wave enters a more dense medium (n1 < n2).
  • The angle of refraction is smaller than the angle of incidence (\thetat < \thetai).

• Outward Refraction:

  • Occurs when a wave enters a less dense medium (n1 > n2).
  • The angle of refraction is larger than the angle of incidence (\thetat > \thetai).

Total Internal Reflection

• Critical Angle: The angle of incidence at which the refraction angle is 90 degrees.

• Total Internal Reflection: When the angle of incidence exceeds the critical angle, the wave is totally reflected back into the original medium.

  • Surface waves exist on the transmission side.
  • The reflection coefficient (\Gamma) equals 1 (\Gamma = 1).

Optical Fiber

• Components:

  • Fiber core (index of refraction n_f).
  • Cladding (index of refraction n_c).
  • Surrounding medium (index of refraction n_0).

• Acceptance Cone: Defines the range of angles within which light can enter the fiber and be guided through it via total internal reflection.

• Waveguiding: Waves can be guided along optical fibers due to successive internal reflections.

Example: Light Beam Passing Through a Slab

• Setup: A dielectric slab (index of refraction n2) surrounded by a medium with index of refraction n1.

• Condition: If the incidence angle (\thetai) is less than the critical angle (\thetac), the emerging beam is parallel to the incident beam.

• Solution:

  • At the upper surface: \frac{\sin \theta2}{\sin \theta1} = \frac{n1}{n2} (8.33)
  • At the lower surface: \frac{\sin \theta3}{\sin \theta2} = \frac{n2}{n1} (8.34)
  • Combining the equations: \sin \theta3 = \frac{n2}{n1} \sin \theta2 = \frac{n2}{n1} \cdot \frac{n1}{n2} \sin \theta1 = \sin \theta1
  • Therefore: \theta3 = \theta1

• Conclusion: The slab displaces the beam's position, but the beam's direction remains unchanged.

Oblique Incidence: Snell's Law

• Inward Refraction:

  • Occurs when n1 < n2.
  • \thetat < \thetai

• Outward Refraction:

  • Occurs when n1 > n2.
  • \thetat > \thetai

EM Polarized Waves

• Traveling and standing waves in different mediums.
• Medium 1: (\mu1, \epsilon1, \sigma1, \eta1, \beta_1)
• Medium 2: (\mu2, \epsilon2, \sigma2, \eta2, \beta_2)

Oblique Incidence

• Plane of Incidence: Defined by the normal to the boundary and the direction of propagation of the incident wave (x-y plane).

Perpendicular/TE Polarization

• Electric field is perpendicular to the plane of incidence.

• Magnetic Field components: Hi, Hr, H_t

• Electric Field components: Ei, Er, E_t

Parallel/TM Polarization

• Electric field is parallel to the plane of incidence.

• Magnetic Field components: Hi, Hr, H_t

• Electric Field components: Ei, Er, E_t

General Polarization

• Electric and Magnetic Field components.
• \vec{k} = k(\cos\phi \hat{x} - \sin\phi \hat{y})
• \vec{r} = x\hat{x} + y\hat{y} + z\hat{z}
• \vec{E} = Ex + Ey
• \vec{H}

Perpendicular Polarization

• Detailed illustration of incident, reflected, and transmitted waves with perpendicular polarization.

• Relationships between electric and magnetic fields, angles, and media properties.

• Incident Wave: \vec{Ei} = \hat{y} E{i0} e^{-jk1(x\sin\thetai + z\cos\thetai)} \vec{Hi} = (-\hat{x} \cos\thetai + \hat{z} \sin\thetai) \frac{E{i0}}{\eta1} e^{-jk1(x\sin\thetai + z\cos\theta_i)}

• Reflected Wave: \vec{Er} = \hat{y} E{r0} e^{-jk1(x\sin\thetar - z\cos\thetar)} \vec{Hr} = (\hat{x} \cos\thetar + \hat{z} \sin\thetar) \frac{E{r0}}{\eta1} e^{-jk1(x\sin\thetar - z\cos\theta_r)}

• Transmitted Wave: \vec{Et} = \hat{y} E{t0} e^{-jk2(x\sin\thetat + z\cos\thetat)} \vec{Ht} = (-\hat{x} \cos\thetat + \hat{z} \sin\thetat) \frac{E{t0}}{\eta2} e^{-jk2(x\sin\thetat + z\cos\theta_t)}

Applying Boundary Conditions

• Tangential E Continuous:

• Tangential H Continuous:

Solution of Boundary Equations

• Exponents must be equal for all values of x.

• Remaining terms become expressions for reflection and transmission coefficients.

Parallel Polarization

• Reflection coefficient: r\parallel = \frac{\eta2 \cos \thetat - \eta1 \cos \thetai}{\eta2 \cos \thetat + \eta1 \cos \theta_i}

• Transmission coefficient: t\parallel = \frac{2\eta2 \cos \thetai}{\eta2 \cos \thetat + \eta1 \cos \theta_i}

• Relationship: t\parallel= (1 + r\parallel) \frac{\cos \thetai}{\cos \thetat} .

General Polarization

• General case with both perpendicular and parallel components.

• \vec{Ei} = \vec{E{\parallel i}} + \vec{E{\perp i}} \vec{Er} = \vec{E{\parallel r}} + \vec{E{\perp r}}
  \vec{Et} = \vec{E{\parallel t}} + \vec{E_{\perp t}}

Example 8-6: Wave Incident Obliquely on a Soil Surface

• A plane wave from a distant antenna is incident on a soil surface at z = 0.

  • \vec{E_i} = \hat{y} 100 \cos( \omega t - \pi x - 1.73\pi z) \quad (V/m)
  • Soil: lossless dielectric with \epsilon_r = 4

• Objectives:

  • Determine k1, k2, and the incidence angle \theta_i.
  • Obtain expressions for the total electric fields in air and soil.
  • Determine the average power density in the soil.

Example 8-6 (cont.)

• 
  [ k r E x y     j x y ˆE ˆE e ~

Example 8-6 (cont.)

• Wavelength in air (medium 1):

  • \lambda_1 = \frac{2\pi}{\pi} = 2 \quad m

• Wavelength in soil (medium 2):

  • \lambda2 = \frac{\lambda1}{\sqrt{\epsilon_{r2}}} = \frac{2}{\sqrt{4}} = 1 \quad m

• Wave number in medium 2:

  • k2 = \frac{2\pi}{\lambda2} = \frac{2\pi}{0.5} = 4\pi \quad (rad/m)

• Incidence angle:

  • \sin \thetai = \frac{\pi}{k1} = \frac{\pi}{2\pi} = 0.5
  • \theta_i = 30^\circ

• Transmission angle:

  • \sin \thetat = \frac{k1}{k2} \sin \thetai = \frac{2\pi}{4\pi} \sin 30^\circ = 0.25
  • \theta_t = 14.5^\circ

• Reflection and transmission coefficients (perpendicular polarization):

  • r\perp = \frac{\cos \thetai - \sqrt{(\epsilon2/\epsilon1) - \sin^2 \thetai}}{\cos \thetai + \sqrt{(\epsilon2/\epsilon1) - \sin^2 \theta_i}} = -0.38
  • t\perp = 1 + r\perp = 0.62

Example 8-6 (cont.)

• Total electric field in medium 1 (air):

  • \vec{E1} = \hat{y} E{i0} e^{-j k1 (x \sin \thetai + z \cos \thetai)} + \hat{y} r\perp E{i0} e^{-j k1 (x \sin \thetai - z \cos \thetai)}
  • \vec{E_1} = \hat{y} 100 e^{-j(\pi x + 1.73 \pi z)} - \hat{y} 38 e^{-j(\pi x - 1.73 \pi z)}

• Instantaneous electric field in medium 1:

  • E1(x, z, t) = Re[\vec{E1} e^{j \omega t}] = \hat{y} [100 \cos(\omega t - \pi x - 1.73 \pi z) - 38 \cos(\omega t - \pi x + 1.73 \pi z)] \quad (V/m)

Example 8-6 (cont.)

• Electric field in medium 2 (soil):

  • \vec{Et} = \hat{y} t\perp E{i0} e^{-j k2 (x \sin \thetat + z \cos \thetat)} = \hat{y} 62 e^{-j(\pi x + 3.87 \pi z)}

• Instantaneous electric field in medium 2:

  • Et(x, z, t) = Re[\vec{Et} e^{j \omega t}] = \hat{y} 62 \cos(\omega t - \pi x - 3.87 \pi z) \quad (V/m)

• Average power density in medium 2:

  • \eta2 = \frac{\eta0}{\sqrt{\epsilon_{r2}}} = \frac{120 \pi}{\sqrt{4}} = 60 \pi \quad (\Omega)
  • S{av} = \frac{|E{t0}|^2}{2 \eta_2} = \frac{(62)^2}{2 \cdot 60 \pi} \approx 10.2 \quad (W/m^2)

Reflection Coefficient vs. Angle

• Plots of the magnitude of the reflection coefficient (\Gamma) for dry soil, wet soil, and water surfaces vs. incidence angle.

• Brewster angle is shown for each surface, where \Gamma_{\parallel} = 0.

Brewster Angle

• Perpendicular Polarization: Does not occur

• Parallel Polarization: Perfect Transmission occurs at Brewster angle (\theta_B).

Summary For Reflection and Transmission

• Table summarizing reflection coefficient (\Gamma), transmission coefficient (t), reflectivity (R), and transmissivity (T) for both normal and oblique incidence, for both perpendicular and parallel polarization.

• Normal Incidence:

  • \thetai = \thetat = 0
  • \Gamma = \frac{\eta2 - \eta1}{\eta2 + \eta1}
  • t = \frac{2 \eta2}{\eta2 + \eta_1}
  • t = 1 + \Gamma
  • R = |\Gamma|^2
  • T = |t|^2 \frac{\eta1}{\eta2}
  • T = 1 - R

• Perpendicular Polarization:

  • \Gamma\perp = \frac{\eta2 \cos \thetai - \eta1 \cos \thetat }{\eta2 \cos \thetai + \eta1 \cos \theta_t}
  • t\perp = \frac{2 \eta2 \cos \thetai}{\eta2 \cos \thetai + \eta1 \cos \theta_t}
  • t\perp = 1 + \Gamma\perp
  • R\perp = |\Gamma\perp|^2
  • T\perp = |t\perp|^2 \frac{\eta1 \cos \thetat }{\eta2 \cos \thetai}
  • T\perp = 1 - R\perp

• Parallel Polarization:

  • \Gamma\parallel = \frac{\eta2 \cos \thetat - \eta1 \cos \thetai}{\eta2 \cos \thetat + \eta1 \cos \theta_i}
  • t\parallel = \frac{2 \eta2 \cos \thetai}{\eta2 \cos \thetat + \eta1 \cos \theta_i}
  • t\parallel = (1 + \Gamma\parallel) \frac{\cos \thetai}{\cos \thetat}
  • R\parallel = |\Gamma\parallel|^2
  • T\parallel = |t\parallel|^2 \frac{\eta1 \cos \thetat}{\eta2 \cos \thetai}
  • T\parallel = 1 - R\parallel