Question

The retardation of a uniaxial negative mineral of thickness 0.03 mm is 5160 nm in its principal section of indicatrix. If the refractive index corresponding to the E-ray is 1.486, the value of the refractive index (correct to three decimal places) of the O-ray is ........

Hint:

The retardation for uniaxial minerals can be calculated using the thickness, refractive indices, and wavelength. The refractive indices for the O-ray and E-ray can be determined from the formula provided.

Answer 1

Correct Answer: 1.658

Step 1: Relate Retardation, Thickness, and Birefringence

The retardation ($\Gamma$) is defined by the difference in path length between the two rays (O-ray and E-ray) traveling through the mineral, and is calculated as:

$$\Gamma = t \cdot |n_o - n_e|$$

Where:

$\Gamma$ = Retardation (path difference)

$t$ = Thickness of the mineral (thin section)

$n_o$ = Refractive index of the O-ray (Ordinary ray)

$n_e$ = Refractive index of the E-ray (Extraordinary ray)

Step 2: Standardize Units

Convert the given thickness ($t$) from millimeters ($\text{mm}$) and the retardation ($\Gamma$) from nanometers ($\text{nm}$) to the same unit, meters ($\text{m}$), for consistency:

Thickness: $t = 0.03 \text{ mm} = 0.03 \times 10^{-3} \text{ m}$

Retardation: $\Gamma = 5160 \text{ nm} = 5160 \times 10^{-9} \text{ m}$

Step 3:  Calculate Birefringence

Rearrange the retardation formula to solve for birefringence ($|n_o - n_e|$):

$$|n_o - n_e| = \frac{\Gamma}{t}$$

Substitute the standardized values:

$$|n_o - n_e| = \frac{5160 \times 10^{-9} \text{ m}}{0.03 \times 10^{-3} \text{ m}}$$

$$|n_o - n_e| = \frac{5160}{0.03} \times 10^{-6} = 172000 \times 10^{-6}$$

$$|n_o - n_e| = 0.172$$

Step 4: Determine $n_o$

The problem states the mineral is uniaxial negative. For a uniaxial negative mineral, the $\text{O-ray}$ has a greater refractive index than the $\text{E-ray}$ ($n_o > n_e$).

Therefore, the birefringence is calculated as:

$$n_o - n_e = 0.172$$

We are given the refractive index of the E-ray: $n_e = 1.486$.

Solve for $n_o$:

$$n_o = n_e + 0.172$$

$$n_o = 1.486 + 0.172$$

$$n_o = 1.658$$

The value of the refractive index of the O-ray is 1.658.