Determination of Refractive Index with Fresnel's Formulas (O7)

Determination of Refractive Index using Fresnel's Formulas (O7)

Objective of the Experiment

• The primary goal is to determine the material-specific refractive index of various types of glass (e.g., Crown, Flint, Heavy Flint glass).

• This is achieved by measuring the angle-dependent reflection coefficient of polarized light at the air/glass interface.

• The determination relies on the application of Fresnel's formulas.

Theoretical Background

Interaction of Light with Matter

• When light interacts with matter, the electromagnetic wave of light causes a displacement of the center of charge of the electrons within an atom from their equilibrium position.

• Based on Coulomb's law, this displacement leads to a restoring, linear force.

• This force can be described as a harmonic oscillation of the electron cloud relative to its equilibrium position.

• The magnitude of this force depends on the charge carrier density and the extent of the charge carrier distribution, which are properties of the atom itself.

Equations for Electron Oscillation

• The force of the electric field ($\vec{E}(t)$) of light on a charge carrier is given by: $\vec{F} = -e \vec{E}(t) = -e \vec{E}_0 \exp(i\omega t)$

• This force can be equated with the equation for harmonic oscillation (neglecting damping in this initial description): $\ddot{\vec{x}}(t) + \omega0^2 \vec{x}(t) = - \frac{1}{m} \vec{F}(t) = - \frac{e}{m} \vec{E}0 \exp(i\omega t)$

  • Here, $m$ is the mass of the electron, and $\omega_0$ is the natural frequency of the system, which can be viewed as an elastic binding.

• By using the ansatz $\vec{x}(t) = \vec{x}<em>0 \exp(i\omega t)$, the solution for this differential equation (DGL) is: $\vec{x}(t) = - \frac{e}{m} \frac{1}{\omega</em>0^2 - \omega^2} \vec{E}(t)$

Induced Dipole Moment and Polarization

• The induced dipole moment $\vec{p}$ of an atom with one electron is defined as the product of charge and displacement:
  

  $\vec{p}<em>i(t) = -e \vec{x}</em>i(t)$

• The polarization $\vec{P}$ of the medium is described by: $\vec{P} = (\varepsilon - 1) \varepsilon_0 \vec{E}$

  • Where $\varepsilon$ is the dielectric constant of the medium and $\varepsilon_0$ is the electric field constant.

• The displacement polarization in the medium is also given by: $\vec{P}(t) = \sum \frac{\vec{p}_i(t)}{V}$

• Considering that there are $N$ atoms per unit volume ($V$) in the material, the polarization becomes:
  

  $\vec{P}(t) = -e \vec{x}(t)N = \frac{e^2N}{m} \frac{1}{\omega<em>0^2 - \omega^2} \vec{E}(t) = (\varepsilon - 1) \varepsilon</em>0 \vec{E}$

Dielectric Constant and Refractive Index

• From the above relationships, the dielectric constant for atoms with a single electron is obtained: $\varepsilon(\omega) = 1 + \frac{e^2N}{\varepsilon<em>0m} \frac{1}{\omega</em>0^2 - \omega^2}$

  • Note: The complex-valued part of $\varepsilon$ is omitted in this derivation due to the neglect of damping.

• According to Maxwell's relation for dilute gases, the relationship between the refractive index ($n$) and the dielectric constant is:
  

  $n^2 = \sqrt{\varepsilon} \approx 1 + \frac{1}{2} (\varepsilon - 1) = 1 + \frac{e^2N}{2\varepsilon<em>0m} \frac{1}{\omega</em>0^2 - \omega^2}$

• For atoms possessing multiple electrons, a summation over the respective oscillator strengths $\text{f}<em>i$ and natural frequencies $\omega</em>{0,i}$ must be performed:
  

  $n^2 = 1 + \frac{e^2N}{2\varepsilon<em>0m} \sum \frac{\text{f}</em>i}{\omega_{0,i}^2 - \omega^2}$

• For concentrated media with low absorption, such as glasses, a correction term is necessary to account for the interaction between dipole moments.

• Conclusion: The refractive index depends on both the atomic properties of the medium and the frequency (or wavelength) of the light.

Light Propagation at an Interface

• Retardation and Wavelength Change: The propagation speed of light in a medium ($c$) is retarded by the refractive index ($n$): $c = c<em>0/n$. Consequently, the wavelength of light is also shortened: $\lambda = \lambda</em>0/n$. ($c<em>0$ and $\lambda</em>0$ are speed and wavelength in vacuum).

• Change in Direction: To fulfill the phase continuity condition at the interface between two media with different refractive indices (see Figure 1, representing reflection and transmission at a plane interface), the direction of propagation in the second medium must change.

• Snell's Law: This condition is satisfied when $\lambda<em>1 \sin \alpha</em>1 = \lambda<em>2 \sin \alpha</em>2$. Here, $\lambda<em>1, \alpha</em>1$ are the wavelength and angle of incidence (relative to the normal) in medium 1, and $\lambda<em>2, \alpha</em>2$ are the corresponding values in medium 2.

  • Using the relation $\lambda<em>{1,2} = \lambda</em>0/n<em>{1,2}$, Snell's Law is derived: $\frac{n</em>1}{n<em>2} = \frac{\sin \alpha</em>2}{\sin \alpha_1} \quad (1)$

Huygens' Principle and Wavefronts

• When an atom is excited by the electromagnetic wave of incident light (denoted by index 'e') with frequency $\omega$), the dipole oscillates at this frequency.

• According to Huygens' principle, this oscillating dipole acts as a source of elementary waves radiating in all directions.

• The superposition of elementary waves from all atoms in the medium results in:

  • A wavefront propagating into the medium (refracted/transmitted wavefront, index 't').

  • A wavefront propagating back in the direction of the incident light (reflected wavefront, index 'r').

Boundary Conditions for Electromagnetic Fields

• At the interface (oriented in x, y directions), the tangential components of the electric and magnetic field strengths must satisfy continuity conditions:

  • $E<em>{e,x} + E</em>{r,x} = E_{t,x}$

  • $B<em>{e,x} + B</em>{r,x} = B_{t,x}$

  • $E<em>{e,y} + E</em>{r,y} = E_{t,y}$

  • $B<em>{e,y} + B</em>{r,y} = B_{t,y}$

Derivation of Reflection Coefficient for TE-polarized Light

• TE-polarization: The electric field is perpendicular to the plane defined by the incident wave vector and the surface normal of the interface ($E_{\perp}$).

• Continuity for tangential E-field component:
  

  $E<em>e + E</em>r - E_t = 0 \quad (2)$

• Continuity for tangential B-field component:
  

  $-B<em>e \cos \alpha</em>{1,e} + B<em>r \cos \alpha</em>{1,r} + B<em>t \cos \alpha</em>{2,t} = 0 \quad (3)$

• Using the law of reflection $\alpha<em>{1,e} = \alpha</em>{1,r}$, the relation $\text{B}<em>i = E</em>i/c<em>i = n</em>i E<em>i/c</em>0$, and equation (2), equation (3) simplifies to:
  

  $E<em>r (n</em>1 \cos \alpha<em>{1,e} + n</em>2 \cos \alpha<em>{2,t}) + E</em>e (n<em>2 \cos \alpha</em>{2,t} - n<em>1 \cos \alpha</em>{1,e}) = 0$

• This leads to the expression for the reflection coefficient $\text{r}$ for perpendicular polarization:
  

  $r<em>{\perp} = \frac{E</em>r}{E<em>e} = \frac{n</em>1 \cos \alpha<em>1 - n</em>2 \cos \alpha<em>2}{n</em>1 \cos \alpha<em>1 + n</em>2 \cos \alpha_2}$

• The intensity of the reflected light corresponds to the square of the E-fields.

Fresnel's Formulas

• Similar calculations for transmission for perpendicular polarization and reflection/transmission for parallel polarized light (E*parallel, or $E_{|}$) yield the following expressions. Here, $\Re{\lbrace…\rbrace}$ denotes the real part and $\Vert…\Vert$ denotes the absolute value of the complex quantity.

• Reflection Coefficient for Parallel Polarization ($R_{|}$):**
  

  $R<em>{|} = \frac{I^{(r)}</em>{|}}{I^{(0)}<em>{|}} = \left|\frac{n</em>2 \cos \alpha<em>1 - n</em>1 \cos \alpha<em>2}{n</em>2 \cos \alpha<em>1 + n</em>1 \cos \alpha<em>2}\right|^2 = \left|\frac{\tan(\alpha</em>1 - \alpha<em>2)}{\tan(\alpha</em>1 + \alpha_2)}\right|^2 \quad (4)$

• Reflection Coefficient for Perpendicular Polarization ($R_{\perp}$):**
  

  $R<em>{\perp} = \frac{I^{(r)}</em>{\perp}}{I^{(0)}<em>{\perp}} = \left|\frac{n</em>1 \cos \alpha<em>1 - n</em>2 \cos \alpha<em>2}{n</em>1 \cos \alpha<em>1 + n</em>2 \cos \alpha<em>2}\right|^2 = \left|\frac{\sin(\alpha</em>1 - \alpha<em>2)}{\sin(\alpha</em>1 + \alpha_2)}\right|^2 \quad (5)$

• Transmission Coefficient for Parallel Polarization ($T_{|}$):**
  

  $T<em>{|} = \left| \frac{n</em>2 \cos \alpha<em>2}{n</em>1 \cos \alpha<em>1} \left( \frac{2n</em>1 \cos \alpha<em>1}{n</em>1 \cos \alpha<em>1 + n</em>2 \cos \alpha<em>2} \right)^2 \right| = \left| \frac{4n</em>1n<em>2 \cos \alpha</em>1 \cos \alpha<em>2}{|n</em>2 \cos \alpha<em>1 + n</em>1 \cos \alpha_2|^2} \right| \quad (6)$

• **Transmission Coefficient for Perpendicular Polarization ($T<em>{\perp}$): $T</em>{\perp} = \left| \frac{4n<em>1n</em>2 \cos \alpha<em>1 \cos \alpha</em>2}{|n<em>1 \cos \alpha</em>1 + n<em>2 \cos \alpha</em>2|^2} \right| \quad (7)$

• Important Note: This formulation is crucial because equation (1) for light incident from an optically denser medium (n1 > n2) does not always yield a real solution for $\alpha<em>2$ for all given values of $n</em>1, n<em>2$ and $\alpha</em>1$. In cases of complex angles $\alpha_2$, total internal reflection (discussed below) occurs.

• Energy Conservation: In non-absorbing media, the principle of energy conservation dictates:
  

  $R<em>{\perp} + T</em>{\perp} = R<em>{|} + T</em>{|} = 1 \quad (8)$

Special Angles of Incidence

Brewster's Angle ($\alpha_B$)

• For parallel-polarized light, there exists an angle of incidence $\alpha<em>1 = \alpha</em>B$ at which the reflection coefficient $R_{|}$ becomes zero (refer to Figure 2, left image).

• From equations (1) and (4), Brewster's angle is derived:
  

  $\tan \alpha<em>B = \frac{n</em>2}{n<em>1} \text{ and } \alpha</em>B + \alpha_2 = \frac{\pi}{2} \quad (9)$

• At Brewster's angle, the reflected and transmitted light directions are perpendicular to each other.

• Since dipoles cannot emit radiation parallel to their oscillation axis, the field component $R_{|}$ is zero.

Critical Angle for Total Internal Reflection ($\alpha_T$)

• When light passes from an optically denser medium (n1 > n2) to an optically thinner medium, complete reflection occurs for angles of incidence \alpha1 > \alphaT (refer to Figure 2, right image).

• The critical angle for total reflection is given by:
  

  $\sin \alpha<em>T = \frac{n</em>2}{n_1} \quad (10)$

• Note: The 'external' (n1 < n2) and 'internal' (n1 > n2) Brewster angles are different.

Experimental Setup and Procedure

Setup Components (Refer to Figure 3)

1. Light Source: Provides the incident light.

2. Pinhole (Lochblende): Shapes the light beam.

3. Lens (Linse): Focuses the light onto the detector.

4. Iris Diaphragm (Irisblende): Adjusts the beam diameter to ensure all light hits the glass block.

5. Polarizer (Polarisator): Sets the polarization state of the light (parallel or perpendicular).

6. IR-Blocking Filter (IR-Sperrfilter): Filters out infrared radiation.

7. Planar Glass Block on Goniometer (Planglasblock auf Goniometer): The test medium. The goniometer allows rotation to change the angle of incidence.

8. Detector (Detektor): A photodiode, whose short-circuit current is proportional to the light intensity. It can be rotated independently around the same axis as the goniometer.

Measurement Procedure

• The experiment uses precisely polished sides of various glass blocks (Crown, Flint, Heavy Flint glass) as the reflecting medium.

• The goniometer is rotated to vary the angle of incidence of the light on the glass block.

• The detector is rotated independently around the same central axis to measure the reflected light.

• The iris diaphragm is adjusted to ensure the entire light beam interacts with the glass block.

• To determine the reflection coefficient, both the reflected light intensity and the incident light intensity must be measured.

Experimental Tasks

1. For three different glass blocks (Crown, Flint, and Heavy Flint glass), determine the dependence of the reflection coefficient on the angle of incidence for both parallel and perpendicularly polarized light.

   • For each measurement, record the background noise (measurement with the light path interrupted) and subtract it from the measured value.

   • Be aware that any change in room lighting conditions impacts the measurement.

   • Compare the collected results with Fresnel's formulas.

2. Determine the refractive indices of the used glass blocks from the measured Brewster angles and compare these values with literature values.

Additional Notes

• Phase Velocity: From Maxwell's equations (for the visible spectral range and non-magnetic materials where permeability $\mu \approx 1$), the phase velocity ($v<em>{Ph}$) is given by: $v</em>{Ph} = \frac{1}{\sqrt{\varepsilon \cdot \mu<em>0}}$ For non-magnetic materials, this simplifies to: $v</em>{Ph} = \frac{1}{\sqrt{\varepsilon \varepsilon<em>0 \mu</em>0}} = \frac{1}{\sqrt{\varepsilon} \frac{1}{c<em>0}} = \frac{c</em>0}{\sqrt{\varepsilon}}$
  

  where $\mu_0$ is the permeability of free space.

• Refractive Index Definition: The refractive index ($n$) of a medium is defined as the ratio of the phase velocity of light in vacuum ($c<em>0$) to that in the medium ($v</em>{Ph}$):
  

  $n = \frac{c<em>0}{v</em>{Ph}} = \sqrt{\varepsilon}$

• Complex Refractive Index: Like $\varepsilon$, $n$ is also generally a complex quantity.

• Transparent Media: For transparent media, which exhibit no absorption (expressed by the complex component of $n$), $n$ can be considered as a real number.